Equality-constrained minimization of polynomial

SCIENCE CHINA Mathematics

. ARTICLES .

October 2015 Vol. 58 No. 10: 2181–2204 doi: 10.1007/s11425-015-5012-6

Equality-constrained minimization of polynomial functions XIAO ShuiJing & ZENG GuangXing∗ Department of Mathematics, Nanchang University, Nanchang 330031, China Email: [email protected], [email protected] Received March 5, 2014; accepted October 11, 2014; published online April 15, 2015

Abstract This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1 , . . . , xn ] the ring of polynomials over R in variables x1 , . . . , xn . For an f ∈ R[x1 , . . . , xn ] and a finite subset H of R[x1 , . . . , xn ], denote by V(f : H) the set {f (¯ α) | α ¯ ∈ Rn , and h(¯ α) = 0, ∀ h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) 6= ±∞. The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients. Keywords polynomial function, equality constraints, equality-constrained minimization, constrained infimum, Wu’s algorithm, triangular decomposition, revised resultant, Transfer principle MSC(2010)

68W30, 12F10, 12J15

Citation: Xiao S J, Zeng G X. Equality-constrained minimization of polynomial functions. Sci China Math, 2015, 58: 2181–2204, doi: 10.1007/s11425-015-5012-6

1

Introduction

The equality-constrained minimization of polynomial functions is an important problem in optimization. This problem can be described as follows. For an f ∈ R[x1 , . . . , xn ] and a finite subset H of R[x1 , . . . , xn ], where R[x1 , . . . , xn ] is the rings of polynomials over the field R of real numbers in variables x1 , . . . , xn , compute the infimum inf V(f : H) of the set V(f : H), where V(f : H) stands for the set {f (¯ α) | α ¯ ∈ Rn , and h(¯ α) = 0, ∀ h ∈ H}. So the equality-constrained minimization is actually the optimization of a polynomial over a real algebraic set. According to the definition of infimums, inf V(f : H) = +∞ if and only if V(f : H) = ∅ (i.e., the feasible region {α ¯ ∈ Rn | h(¯ α) = 0, ∀ h ∈ H} is empty), and inf V(f : H) = −∞ if and only if V(f : H) is nonempty and is not bounded from below. In the special case when H = ∅, the above-mentioned problem is just the global minimization of polynomial functions. The global minimization of polynomial functions was investigated by many researchers, see for example [6, 7, 10, 12, 16, 17, 23, 30, 31, 33]. In the general case when H 6= ∅, the classical method of Lagrange multipliers is an important theoretical tool for the equality-constrained minimization of polynomial functions. However, the method of Lagrange multipliers replies on the assumptions that inf V(f : H) is attained and the rank of the Jacobian matrix ∗ Corresponding

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for polynomials in H is equal to the number of polynomials in H, see [22, Theorem 5.1]. It is often that a desired result is not leaded by the Lagrange’s method, because the solutions of the system of equations, which are induced by the Theorem of Lagrange, cannot be simply computed. Early in 1992, Wu investigated the equality-constrained minimization of polynomial functions based on the Lagrange’s theorem and his Zero-Decomposition Theorem II, see [27] or [28, Subsection 5.5]. According to the finite Kernel theorem in [27, 28] and its proof, there is an algorithm which gives a finite number of univariate polynomials such that inf V(f : H) is a root of some univariate polynomial when inf V(f : H) is attained. Let f ∈ R[x1 , . . . , xn ] and let H be a finite subset of R[x1 , . . . , xn ] such that inf V(f : H) is attained. By introducing an auxiliary variable t, the design of this algorithm is based on such a fact as follows: If a triangular decomposition of the set {t − f } ∪ H of polynomials with respect to the lexicographic order t ≺ x1 ≺ · · · ≺ xn is computed according to the Zero-Decomposition Theorem II, then inf V(f : H) is a root of certain polynomial in one variable t, which is contained in some ascending chain of the triangular decomposition. In the case when inf V(f : H) is attained, Nie [14, 15] utilized the Semidefinite programming (SDP) relaxations in dealing with the constrained minimization of polynomial functions. The validity of SDP relaxations is based on the Putinar’s Positivstellensatz [19]. Hence, the method of SDP relaxations requires certain assumptions on the feasible region in the optimization problem. H´ a and Pham [5] proposed a method for finding the infimum of a polynomial f on a basic closed semi-algebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case when the polynomial f does not attain its infimum on S. The method in [5] requires the assumption that S is regular, although basic closed semi-algebraic sets are more general than the feasible regions in the equality-constrained minimization. Greuet et al. [3] investigated by SDP relaxations the equality-constrained minimization without assumption that the infimum is attained. However, the following assumptions are required in [3]: (i) The ideal generated by the constraint polynomials is radical and equidimensional; (ii) The complex zero set of the constraint polynomials is smooth. Recently, Greuet and Safey El Din [4] gave a probabilistic and exact algorithm for the equality-constrained minimization. The algorithm in [4] is also adapted to the case when the desired infimum does not attain, but certain assumptions analogue to those in [3] are required. The purpose of this paper is to present a new algorithm for the above-mentioned equality-constrained minimization of polynomial functions. As in [27], our algorithm also yields a finite set U of non-zero univariate polynomials such that the constrained infimum inf V(f : H) is a root of some polynomial in U whenever inf V(f : H) 6= ±∞. The strategies of this paper are decomposing a polynomial system (i.e., a finite set of polynomials) into triangular chains of polynomials and computing the so-called revised resultants. So the efficiency of our algorithm relies mainly on the triangular decompositions of polynomial systems. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients. Compared with the method in [27] or in [28, Subsection 5.5], the algorithm in this paper possesses the following features: (1) There is no assumption that inf V(f : H) is attained. Hence the desired result may be obtained whether inf V(f : H) is attained or not. (2) The notion of revised resultants is introduced to raise the efficiency of computation. (3) In the process of decomposing a polynomial system into triangular chains, such a polynomial t − f is not involved, where f is the objective polynomial function and t is an auxiliary variable. The remainder of this paper is divided into four sections. As some technical preliminaries, Section 2 consists of two subsections. In Subsection 2.1, we recall some basic concepts on the triangular decompositions of polynomial systems, and establish some results on the regular zeros of a triangular chain of polynomials. In Subsection 2.2, we introduce the so-called revised resultant for a non-zero polynomial and a chain of polynomials, and establish some useful results on revised resultants. In Section 3, we establish an algorithm for catching the constrained infimum of a polynomial. For an f ∈ F[x1 , . . . , xn ] and a finite subset H of F[x1 , . . . , xn ], where F is a computable ordered subfield of R, this algorithm can yield a finite set U of non-zero univariate polynomials such that the constrained infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) 6= ±∞. According to inf V(f : H) being attained or

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not, we consider the special case and the general case respectively, and establish some theoretic results related to our algorithms. In the general case when there is no assumption that inf V(f : H) is attained, the field R of real numbers is extended to a real closed field containing infinitely large elements. In Section 4, two examples are given to illustrate our algorithms. The final section is the conclusion of this paper. Throughout this paper, F stands for a computable subfield of the field R of real numbers, and F[x1 , . . . , xn ] the ring of polynomials over F in variables x1 , . . . , xn . In the ring F [x1 , . . . , xn ] of polynomials over any field F in variables x1 , . . . , xn , the involved lexicographic order is assumed to be the natural order x1 ≺ · · · ≺ xn in default of lexicographic orders.

2

Preliminaries

Before establishing the main results and algorithms, we need some technical preliminaries. Our discussion will proceed in two subsections. 2.1

Triangular chains of polynomials and their regular zeros

In this subsection, we recall some basic concepts and results on the triangular decompositions of polynomial systems. Moreover, we introduce the notion of regular zeros for triangular chains of polynomials, and establish some results on regular zeros. The triangular decomposition of a polynomial system was introduced by Ritt [20]. Ritt’s decomposition relies on computing the so-called characteristic sets, which are some triangular sets of polynomials, of prime ideals, see [21]. So Ritt’s decomposition involves the factorization of polynomials in algebraic extensions. In order to avoid such a factorization of polynomials, Wu [26] provided an algorithm for solving polynomial systems by means of characteristic sets of not necessarily prime ideals. For the details of Wu’s algorithm and its applications, refer to [28]. Let F be an arbitrary field, let F [x1 , . . . , xn ] be the ring of polynomials over F in variables x1 , . . . , xn , and xj1 , . . . , xjn a permutation of the variables x1 , . . . , xn . For a non-constant polynomial f ∈ F [x1 , . . . , xn ], a variable xji (1 6 i 6 n) is called the main variable of f with respect to the lexicographic order xj1 ≺ · · · ≺ xjn , if f ∈ F [xj1 , . . . , xji ] but f ∈ / F [xj1 , . . . , xji−1 ]. The main variable xji of f is denoted by mv(f ), and the leading coefficient of f , as a polynomial over K[xj1 , . . . , xji−1 ] in one variable xji , is ∂f called the initial of f . Moreover, the partial derivative ∂x of f with respect to the main variable xji ji is called the separant of f . A sequence C := [f1 , . . . , fs ] of non-constant polynomials in F [x1 , . . . , xn ] is called a chain (ascending chain or triangular set) with respect to the lexicographic order xj1 ≺ · · · ≺ xjn , if mv(f1 ) ≺ · · · ≺ mv(fs ). According to [28, Definition 3.20], a chain [f1 , . . . , fs ] in F [x1 , . . . , xn ] (with respect to a lexicographic order) is called complete (or incomplete) if s = n (or s < n). Now let P and Q be two finite subsets of F [x1 , . . . , xn ], and E an arbitrary extension of F . Then we may obtain such a subset of E n as follows: ZeroE (P/Q) = {α ¯ ∈ E n | p(¯ α) = 0 for all p ∈ P, but q(¯ α) 6= 0 for all q ∈ Q}. In particular, when Q = {1}, we prefer using ZeroE (P ) instead of ZeroE (P/{1}). As two main results on his algorithms, Wu established the so-called Zero-Decomposition Theorems I and II. According to [28, Subsection 3.5, Theorems 5.1 and 5.2], these Zero-Decomposition theorems may be cited as follows: Zero-Decomposition Theorem I. For a finite subset P of F [x1 , . . . , xn ] where F is a computable field, a set C1 , . . . , Cr of chains in F [x1 , . . . , xn ] can be effectively obtained such that for an arbitrary extension E of F , the following equality holds: [ ZeroE (P ) = ZeroE (Ci /Ii ), 16i6r

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where Ii is the initial set of Ci (i.e., the set of the initials of members in Ci ), i = 1, . . . , r. Zero-Decomposition Theorem II. For a finite subset P of F [x1 , . . . , xn ] where F is a computable field, a set C1 , . . . , Cr of chains in F [x1 , . . . , xn ] can be effectively obtained such that for an arbitrary extension E of F , the following equality holds: [ ZeroE (P ) = ZeroE (Ci /Qi ), 16i6r

where Qi is the initial-separant set of Ci (i.e., the set of the initials and separants of members in Ci ), i = 1, . . . , r. In view of these Zero-Decomposition theorems, we give the following definition. Definition 2.1. Let P be a finite subset of F [x1 , . . . , xn ] where F is a field, and let C1 , . . . , Cr be a set of chains in F [x1 , . . . , xn ]. C1 , . . . , Cr is called a triangular decomposition of P if for an arbitrary S extension E of F , ZeroE (P ) = 16i6r ZeroE (Ci /Ii ), where Ii is the initial set of Ci , i = 1, . . . , r. C1 , . . . , Cr is called a triangular decomposition of P with separants, if for an arbitrary extension E of F , S ZeroE (P ) = 16i6r ZeroE (Ci /Qi ), where Qi is the initial-separant set of Ci , i = 1, . . . , r. It should be pointed out that the algorithms related to the above Zero-Decomposition Theorems have been made into a computer software named wsolve. The software wsolve contains two important functions wsolve and evalue (old name: e val) for creating triangular decompositions and triangular decompositions with separants respectively. The software wsolve can be found in [24]. Definition 2.2. Let C := [f1 , . . . , fs ] be a chain in F [x1 , . . . , xn ] with initial set I and main variables xj1 ≺ · · · ≺ xjs , where 1 6 j1 < · · · < js = n, put Tv(C) := {x1 , . . . , xn } \ {xjk | k = 1, . . . , s}, and write ΩC for the algebraic closure of F (Tv(C)), the fraction field of the polynomial ring F [Tv(C)]. A zero ξ¯ := (ξ1 , . . . , ξn ) in ZeroΩC (C/I) is called a regular zero of C if the following conditions are satisfied: (1) For i ∈ {1, . . . , n}, ξi = xi whenever xi ∈ Tv(C). (2) For k ∈ {1, . . . , s}, ξjk is a root of the univariate polynomial fk (ξ1 , . . . , ξjk −1 , xjk ) in ΩC . In what follows, denote by RZ(C) the set of all regular zeros of C. ¯ = f for Remark 2.3. It is easy to see that RZ(C) is finite for any chain C in F [x1 , . . . , xn ] and f (ξ) ¯ any polynomial f ∈ F [Tv(C)] and any ξ ∈ RZ(C). Example.

Let Q be the rational numbers, and let C1 := {x21 − x22 , (x2 − x1 )x3 },

C2 := {(x21 − x22 )2 , (x2 − x1 )x3 },

C3 := {x1 − x2 , (x2 − x1 )x3 },

C4 := {(x21 − x22 )2 , x2 (x3 − x1 )}.

Obviously, these sets are chains with respect to the lexicographic order x1 ≺ x2 ≺ x3 . According to Definition 2.2, we have RZ(C1 ) = {(x1 , −x1 , 0)}, RZ(C3 ) = ∅,

RZ(C2 ) := {(x1 , −x1 , 0)},

RZ(C4 ) := {(x1 , x1 , x1 ), (x1 , −x1 , x1 )}.

Lemma 2.4. Let C := [g1 , . . . , gs ] be a chain in F [x1 , . . . , xn ], and let Q be a finite subset of F [x1 , . . . , xn ] containing the initial set of C. If α ¯ ∈ ZeroK (C/Q) where K is a field extension of F , ¯ then there exists a regular zero ξ of C such that the following condition is satisfied: For h ∈ F [x1 , . . . , xn ], ¯ = 0. h(¯ α) = 0 whenever h(ξ) Proof. Denote by w the product of polynomials in Q, and write J for the ideal in the polynomial ring F [y, x1 , . . . , xn ] generated by 1 − yw, g1 , . . . , and gs , where y is a new variable. Suppose that 1 ∈ J. Then there exist v, v1 , . . . , vs in F [y, x1 , . . . , xn ] such that 1 = v(1 − yw) + v1 g1 + · · · + vs gs . Since α ¯ ∈ ZeroK (C/Q), we have w(¯ α) 6= 0. By the substitution that (x1 , . . . , xn ) = α ¯ and y = 1/w(¯ α) we get 1 = 0, a contradiction. Thus 1 ∈ / J, i.e., J is a proper ideal of F [y, x1 , . . . , xn ].

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√ √ Denote√by J the radical of J. Since F [y, x1 , . . . , xn ] is a Noetherian ring, J can be represented in Tm the form J = i=1 ℘i , where ℘i is prime ideal of F [y, x1 , . . . , xn ] for i = 1, . . . , m. By [11, Chapter 7, Corollary 3.6], dim(℘i ) > (n + 1) − (s + 1) = n − s, where dim(℘i ) stands for the (Krull) dimension of ℘i and i = 1, . . . , m. √ Sm α, 1/w(¯ α)) Observe that (¯ α, 1/w(¯ α)) ∈ ZeroK (J) = ZeroK ( J) = i=1 ZeroK (℘i ). It follows that (¯ ∈ ZeroK (℘) for some ℘ ∈ {℘1 , . . . , ℘m }. Denote by Ω the algebraic closure of the fraction field of the domain F [y, x1 , . . . , xn ]/℘, and put βi := xi + ℘ for i = 1, . . . , n. Obviously, (β1 , . . . , βn , 1/w(β1 , . . . , βn )) ∈ ZeroΩ (℘) ⊆ ZeroΩ (J). Put d := dim(℘). Then d > n−s, and the transcendence degree of F (β1 , . . . , βn ) (= F (β1 , . . . , βn , 1/w(β1 , . . . , βn )) over F is just d. Assume that xj1 ≺ · · · ≺ xjs are the main variables of C, where 1 6 j1 < · · · < js = n. Then βjk is a root of the non-zero univariate polynomial fk (β1 , . . . , βjk −1 , xjk ) in F (β1 , . . . , βn ) for k ∈ {1, . . . , s}. This implies that F (β1 , . . . , βn ) is an algebraic extension of F (β1 , . . . , βj1 −1 , . . . , βjs−1 +1 , . . . , βjs −1 ). Hence, the transcendency degree of F (β1 , . . . , βj1 −1 , . . . , βjs−1 +1 , . . . , βjs −1 ) over F also is d. So we have d 6 n − s and d = n − s. Thus β1 , . . . , βj1 −1 , . . . , βjs−1 +1 , . . . , βjs −1 are algebraically independent over F , and there is an F -isomorphism σ of the field F (β1 , . . . , βj1 −1 , . . . , βjs−1 +1 , . . . , βjs −1 ) onto F (x1 , . . . , xj1 −1 , . . . , xjs−1 +1 , . . . , xjs −1 ) such that (β1 , . . . , βj1 −1 , . . . , βjs−1 +1 , . . . , βjs −1 ) 7→ (x1 , . . . , xj1 −1 , . . . , xjs−1 +1 , . . . , xjs −1 ). Observe that Ω and ΩC are the algebraic closures of F (β1 , . . . , βj1 −1 , . . . , βjs−1 +1 , . . . , βjs −1 ) and F (x1 , . . . , xj1 −1 , . . . , xjs−1 +1 , . . . , xjs −1 ), respectively. So σ can be extended to an F -isomorphism, denoted still by σ, of Ω onto ΩC . Put ξi := σ(βi ) for i = 1, . . . , n. According to Definition 2.2, it is easy to see that ξ¯ := (ξ1 , . . . , ξn ) ∈ RZ(C). ¯ = 0. Then it follows that σ(h(β1 , . . . , βn )) Now let h ∈ F [x1 , . . . , xn ] be a polynomial such that h(ξ) ¯ = h(ξ) = 0. So we have h(β1 , . . . , βn ) = 0, i.e., h ∈ ℘. Observing that (¯ α, 1/w(¯ α)) ∈ ZeroK (℘), we get h(¯ α) = h(¯ α, 1/w(¯ α)) = 0. This completes the proof. By Lemma 2.4 and its proof, the following corollary can be immediately established. Corollary 2.5. Let C := [g1 , . . . , gs ] be a chain in F [x1 , . . . , xn ], and Fb the algebraic closure of F . Then following statements are equivalent: (1) RZ(C) 6= ∅; (2) ZeroK (C/I) 6= ∅ for some extension K of F ; (3) ZeroFb (C/I) 6= ∅. 2.2

Revised resultants

In this subsection, we introduce the so-called revised resultant for a non-zero polynomial and a chain of polynomials, and establish some useful results on revised resultants. Firstly, we give the following definition. Definition 2.6. Let F be a field, and let f , g ∈ F [x1 , . . . , xn ] be two non-zero polynomials. A polynomial m in F [x1 , . . . , xn ] is called a greatest divisor of f coprime to g if the following conditions are satisfied: (1) m is a divisor of f , and m and g are coprime. (2) If h is a divisor of f and h and g are coprime, then h divides m. (3) The leading coefficient of m is the same as f with respect to the lexicographic order x1 ≺ · · · ≺ xn . For any pair of non-zero polynomials f , g in F [x1 , . . . , xn ], it is easy to prove that there exists a greatest divisor of f coprime to g and it is uniquely determined by f and g. Hence, such a divisor m as in Definition 2.6 should be called the greatest divisor of f coprime to g. The following algorithm may be used in computing such greatest divisors as in Definition 2.6, if F is a computable field allowing an effective method of computing greatest common divisors of non-zero polynomials in F [x1 , . . . , xn ], for example F = Q the field of rational numbers. For two non-zero polynomials f , g in F [x1 , . . . , xn ], in the following algorithm, gcd(f, g) stands for the greatest common divisor of f and g having leading coefficient 1 with respect to the lexicographic order x1 ≺ · · · ≺ xn .

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Algorithm 1 (Finding the greatest divisor). Structure: A computable field F allowing an effective method of computing greatest common divisors of non-zero polynomials in F [x1 , . . . , xn ]. Input: Two non-zero polynomials f , g in F [x1 , . . . , xn ]. Output: The greatest divisor of f coprime to g. Procedure: Compute recursively a pair (m, h) of polynomials in F [x1 , . . . , xn ] as follows: • m := f and h := gcd(f, g); • while h 6= 1 do m := m h , and h := gcd(m, g); • RETURN(m). Let f , g be two non-zero polynomials in F [x1 , . . . , xn ]. For a fixed variable xi with 1 6 i 6 n, both f and g may be regarded as polynomials over the ring F [x1 , . . . , xi−1 , xi+1 , . . . , xn ] in one variable xi . According to the definition of resultants (see [1, Notation 4.22] or [13, Definition 7.2.2]), we may obtain the resultant Res(f, g; xi ) of f and g relative to xi . This resultant Res(f, g; xi ) is a polynomial in F [x1 , . . . , xi−1 , xi+1 , . . . , xn ] such that Res(f, g; xi ) = 0 if and only if f and g have a common divisor in which the variable xi really appears. With the aid of the definition of resultants, we give the following definition. Definition 2.7. Let f , g be two non-zero polynomials in F [x1 , . . . , xn ], and m the greatest divisor of f coprime to g. For i = 1, . . . , n, the resultant of m and g relative to xi is called the revised resultant of f and g relative to xi , and is denoted by RvRes(f, g; xi ). Since m and g are coprime, the revised resultant RvRes(f, g; xi ) is a non-zero polynomial in F [x1 , . . . , xi−1 , xi+1 , . . . , xn ] even if the resultant of f and g relative to xi is equal to 0. Now let C := [g1 , . . . , gs ] be a chain in F [x1 , . . . , xn ], and f a non-zero polynomial in F [t, x1 , . . . , xn ] where t is a new variable. Regarding both f and the members in C as polynomials over the ring F [t] in variables x1 , . . . , xn , we get successively the revised resultants as follows: rs−1 := RvRes(f, gs ; mv(gs )), rs−2 := RvRes(rs−1 , gs−1 ; mv(gs−1 )), .. . r1 := RvRes(r2 , g2 ; mv(g2 )), r0 := RvRes(r1 , f1 ; mv(g1 )). Obviously, r0 is a non-zero polynomial in F [t, Tv(C)]. So r0 may be considered as a polynomial over the ring F [Tv(C)] (hence over the field ΩC ) in one variable t. Definition 2.8. Let C and f be as above. The polynomial r0 is called the revised resultant of f with respect to C, and is denoted by RvRes(f ; C) or RvRes(f ; g1 , . . . , gs ). As an useful result on regular zeros and revised resultants, we may establish the following proposition. Proposition 2.9. Let C be a chain in F [x1 , . . . , xn ], and f ∈ F [x1 , . . . , xn ]. Then, for every ξ¯ ∈ ¯ RZ(C), t − f (ξ) is a divisor of RvRes(t − f ; C) as two polynomials over ΩC in one variable t. Proof. We proceed by induction on the number of members in C. Assume that C := [g] and ξ¯ ∈ RZ(C). Without loss of generality, we further assume mv(g) = xn . Then Tv(C) = {x1 , . . . , xn−1 }. Regard g as a polynomial over ΩC in one variable xn , and assume that α1 , . . . , αd are all the root of g in ΩC where d := deg(g, xn ). By Definition 2.2, it is obvious that RZ(C) = {(x1 , . . . , xn−1 , αk ) | k = 1, . . . , d}. Hence ξ¯ = (x1 , . . . , xn−1 , αj ) for some j ∈ {1, . . . , d}. Observe that t − f and g are coprime. Then RvRes(t − f ; C) = RvRes(t − f, g; xn ) = Res(t − f, g; xn ). By a familiar fact about resultants (see [1, Theorem 4.26]), we have RvRes(t − f ; C) = (−1)rd Lr

d Y

k=1

(t − f (x1 , . . . , xn−1 , αk )),

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where L is the initial of g, and r := deg(t − f, xn ). ¯ (i.e., t − Observe that (−1)rd Lr ∈ F [Tv(C)] ⊂ ΩC . The above equality implies that t − f (ξ) ¯ f (x1 , . . . , xn−1 , αj )) is a divisor of RvRes(t − f ; C) where both t − f (ξ) and RvRes(t − f ; C) are regarded as polynomials over ΩC in one variable t. Now assume that the proposition is true for all chains of s − 1 polynomials in F [x1 , . . . , xn ], where s is a positive integer and s > 1. Now let C := [g1 , . . . , gs ] be any chain in F [x1 , . . . , xn ], and ξ¯ := (ξ1 , . . . , ξn ) ∈ RZ(C). Then ξ¯ ∈ ZeroΩC (C/I), where I is the initial set of C, because RZ(C) ⊆ ZeroΩC (C/I). Put C1 := [g2 , . . . , gs ]. Then C1 is a chain of s − 1 polynomials in F [x1 , . . . , xn ]. Obviously, ξ¯ ∈ ZeroΩC (C/I) ⊆ ZeroΩC1 (C1 /I1 ), where I1 is the initial set of C1 . By Lemma 2.4, there exists a µ ¯ := (µ1 , . . . , µn ) ∈ RZ(C1 ) such that the ¯ = 0 whenever h(¯ following statement is true: For h ∈ F [x1 , . . . , xn ], h(ξ) µ) = 0. By the assumption that the proposition is true for all chains of s − 1 polynomials in F [x1 , . . . , xn ], t − f (¯ µ) is a divisor of RvRes(t − f ; C1 ). Obviously, f (¯ µ) is algebraic over F (Tv(C1 )). Hence, there is an irreducible polynomial u(t, Tv(C1 )) in the polynomial ring F [t, Tv(C1 )] such that u(f (¯ µ), Tv(C1 )) = 0. Observe that F [t, Tv(C1 )] is a unique factorization domain and u(t, Tv(C1 )), RvRes(t−f ; C1 ) have a common root f (¯ µ) as two polynomials over F [Tv(C1 )] in one variable t. By the irreducibility of u(t, Tv(C1 )), u(t, Tv(C1 ), t) is a divisor of RvRes(t−f ; C1 ) as two polynomials in F [t, Tv(C1 )]. Hence, RvRes(t−f ; C1 ) = u(t, Tv(C1 ))v(t, Tv(C1 )) for some v(t, Tv(C1 )) ∈ F [t, Tv(C1 )]. Obviously, u(t, Tv(C1 )) is not a divisor of g1 as polynomials in F [t, Tv(C1 )]. By the irreducibility of u(t, Tv(C1 )), it is easy to see that u(t, Tv(C1 )) and g1 are coprime. Denote by w(t, Tv(C1 )) the greatest divisor of v(t, Tv(C1 )) coprime to g1 . Then, it is easy to see that u(t, Tv(C1 ))w(t, Tv(C1 )) is the greatest divisor of RvRes(t − f ; C1 ) coprime to g1 . By Definitions 2.7 and 2.8, we have RvRes(t − f ; C) = Res(u(t, Tv(C1 ))w(t, Tv(C1 )), g1 ; mv(g1 )). Write xj1 for the main variable mv(g1 ) of g1 , where 1 6 j1 < n, and assume that β1 , . . . , βλ are all the root of g1 , as a polynomial over F (x1 , . . . , xj1 −1 ) in one variable xj1 , in ΩC , where λ := deg(g1 , xj1 ). By the definition of RZ(C), it is obvious that ξj1 = βℓ for some ℓ ∈ {1, . . . , λ}. By a familiar fact about resultants (see [1, Theorem 4.26]), we have RvRes(t − f ; C) = Res(u(t, Tv(C1 ))w(t, Tv(C1 )), g1 ; xj1 ) = (−1)mλ Lr1

λ Y

u(t, x1 , . . . , xj1 −1 , βk )w(t, x1 , . . . , xj1 −1 , βk ),

k=1

where L1 is the initial of g1 , and m := deg(u(t, Tv(C1 ))w(t, Tv(C1 )), xj1 ). ¯ in the ring F [ξ1 , . . . , ξn ]. If h1 (¯ For h ∈ F [x1 , . . . , xn ], we define φ(h(¯ µ)) to be the element h(ξ) µ) = ¯ − h2 (¯ µ) for h1 , h2 ∈ F [x1 , . . . , xn ], then h1 (¯ µ) − h2 (¯ µ) = 0. By the preceding statement, we get h1 (ξ) ¯ ¯ ¯ h2 (ξ) = 0, i.e., h1 (ξ) = h2 (ξ). So φ is a mapping of F [µ1 , . . . , µn ] into F [ξ1 , . . . , ξn ]. It is easy to see that φ is an F -homomorphism of F [µ1 , . . . , µn ] into F [ξ1 , . . . , ξn ]. Obviously, φ can be uniquely extended to a ring homomorphism, denoted still by φ, of F [µ1 , . . . , µn ][t] into F [ξ1 , . . . , ξn ][t] such that t 7→ t. Observe that u(t, Tv(C1 )) ∈ F [Tv(C1 ), t] ⊆ F [µ1 , . . . , µn ][t] and f (¯ µ) is a root of u(t, Tv(C1 )) in F [µ1 , . . . , µn ]. ¯ Then it is easy to see that f (ξ) = (φ(f (¯ µ))) is a root of φ(u(t, Tv(C1 ))) in F [ξ1 , . . . , ξn ]. By Definition 2.2, we have Tv(C1 ) = Tv(C) ∪ {xj1 }. Thereby φ(x) = x for all x ∈ Tv(C), but φ(xj1 ) = φ(µj1 ) = ξj1 = βℓ . ¯ = (φ(f (¯ It follows that φ(u(t, Tv(C1 ))) = u(t, x1 , . . . , xj1 −1 , βℓ ). In other words, f (ξ) µ))) is a root of u(t, x1 , . . . , xj1 −1 , βℓ ). ¯ is a root of RvRes(t − f ; C), i.e., t − f (ξ) ¯ is a divisor of RvRes(t According to the above equality, f (ξ) − f ; C). Thus, the proposition also is true for all chains of s polynomials in F [x1 , . . . , xn ]. This completes the proof. Let P be a finite subsets of F [x1 , . . . , xn ], and denote by Id(P ) the ideal of F [x1 , . . . , xn ] generated by P . For a polynomial Π in F [x1 , . . . , xn ], such a subset Id(P ) : Π of F [x1 , . . . , xn ] may be defined as follows: Id(P ) : Π := {f ∈ F [x1 , . . . , xn ] | Πk · f ∈ Id(P ) for some positive integer k}.

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It is easy to prove that Id(P ) : Π is an ideal of F [x1 , . . . , xn ] containing Id(P ). In what follows, Id(P ) : Π is called the quotient ideal of Id(P ) by Π. Now let Q be another finite subset of F [x1 , . . . , xn ], and write ΠQ for the product of all polynomials in Q. So we obtain the quotient ideal Id(P ) : ΠQ of Id(P ) by ΠQ . As usual, Id(P ) : ΠQ is called the quotient ideal of Id(P ) by Q and is simply denoted by Id(P ) : Q. Now we proceed to establish the following theorem, which is a cornerstone for the main algorithm in this paper. Theorem 2.10. Let P and Q be two finite subsets of F [x1 , . . . , xn ], and let f be a polynomial in F [x1 , . . . , xn ] such that (Id(P ∪ {t − f }) : Q) ∩ F [t] 6= {0}, where t is a new variable. If {C1 , . . . , Cs } is a triangular decomposition of P and ui (t) stands for the greatest common divisor of all coefficients of RvRes(t − f, Ci ) as a polynomial over F [t] in variables Tv(Ci ), i = 1, . . . , s, then f (¯ α) is a root of some polynomial among u1 (t), . . . , us (t) for any field extension E of F and any α ¯ ∈ ZeroE (P/Q). Proof. Since (Id(P ∪ {t − f }) : Q) ∩ F [t] 6= {0}, there is a non-zero polynomial h(t) in (Id(P ∪ {t − f }) : Q)∩F [t]. As in the above, write ΠQ for the product of all polynomials in Q. Then ΠkQ ·h ∈ Id(P ∪{t−f }) for some positive integer k. Putting P := {p1 , . . . , pm }, we have ΠkQ · h = w1 (t − f ) + w1 p1 + · · · + wm pm , where w0 , w1 , . . . , wm ∈ F [t, x1 , . . . , xn ]. Let E be any field extension of F , and α ¯ ∈ ZeroE (P/Q). Since {C1 , . . . , Cs } is a triangular decomposition of P , we have [ α ¯ ∈ ZeroE (P/Q) ⊆ ZeroE (P ) = ZeroE (Ci /Ii ). 16i6s

Then there exists a j ∈ {1, . . . , s} such that α ¯ ∈ ZeroE (Cj /Ij ) where Ij is the initial set of Cj . By Lemma 2.4, there is a regular zero ξ¯ ∈ RZ(Cj ) such that the following condition is satisfied: For h ∈ ¯ = 0. F [x1 , . . . , xn ], h(¯ α) = 0 whenever h(ξ) ¯ Observe that ξ ∈ RZ(Cj ) ∈ ZeroΩCj (Cj /Ij ) ⊆ ZeroΩCj (P ). By the substitution that (x1 , . . . , xn ) = ξ¯ ¯ in the above equality, we get ΠQ (ξ) ¯ k · h(f (ξ)) ¯ = 0. Observe that ΠQ (¯ and t = f (ξ) α) 6= 0 as α ¯ ∈ ¯ ¯ ZeroE (P/Q). By the above-mentioned condition, we have ΠQ (ξ) 6= 0. It follows that h(f (ξ)) = 0. Thus ¯ is algebraic over F . Denote by D(t) the minimal polynomial of f (ξ) ¯ over F . Then D(t) is irreducible f (ξ) ¯ also is a root of RvRes(t − f ; Cj ) as a polynomial in F [t]; hence in F [t, Tv(Cj )]. By Proposition 2.9, f (ξ) over ΩCj in one variable t. Hence D(t) and RvRes(t − f ; Cj ) are not coprime. By the irreducibility of D(t), D(t) is a divisor of RvRes(t − f ; Cj ). In other words, D(t) is a common divisor of all coefficients of RvRes(f, Cj ) as a polynomial over F [t] in variables Tv(Cj ). This implies that D(t) is a divisor of uj (t), because uj (t) is the greatest common divisor of all coefficients of RvRes(f, Cj ) as a polynomial ¯ = 0, we get uj (f (ξ)) ¯ = 0. According to the over F [t] in variables Tv(Cj ). Observing that D(f (ξ)) above-mentioned condition, we have uj (f (¯ α)) = 0. This completes the proof. In views of Theorem 2.10, we give the following definition. Definition 2.11. Let f ∈ F [x1 , . . . , xn ], and C a chain in F [x1 , . . . , xn ]. The following procedure of computing is called the algorithm CUP (the algorithm for creating univariate polynomial) adapted to the pair (f, C): Step 1. Introduce a new variable t, and compute RvRes(t − f, C). Step 2. Extract all the coefficients of RvRes(t − f, C) as a polynomial over F [t] in variables Tv(C), and compute their greatest common divisor u(t). Step 3. RETURN(u(t)). By Definition 2.11, Theorem 2.10 can be described in the following version. Theorem 2.12. Let f ∈ F [x1 , . . . , xn ], let P be a finite subset of F [x1 , . . . , xn ] with triangular decomposition C1 , . . . , Cs , and let ui (t) be the output of the algorithm CUP adapted to (f, Ci ) for i = 1, . . . , s. If Q is a finite subset of F [x1 , . . . , xn ] such that (Id(P ∪ {t − f }) : Q) ∩ F [t] 6= {0}, then f (¯ α) is a root of some polynomial among u1 (t), . . . , us (t) for any field extension E of F and any α ¯ ∈ ZeroE (P/Q).

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Catching the constrained infimum of a polynomial

In this section, we establish an algorithm for catching the constrained infimum of a polynomial. For an f ∈ R[x1 , . . . , xn ] and a finite subset H of R[x1 , . . . , xn ], this algorithm can yield a finite set U of non-zero univariate polynomials such that the constrained infimum inf V(f : H) is a root of some polynomial in U whenever inf V(f : H) 6= ±∞. According to inf V(f : H) is attained or not, our discussion proceeds respectively in the special case and the general case. 3.1

The special case when inf V(f : H) is attained

In this subsection, we consider the special case when inf V(f : H) is attained. For generality, we discuss the equality-constrained minimization of polynomials over a real closed field. Let R be a real closed field, let K be a subfield of R, and 6 the only ordering of R. For the sake of convenience, we introduce the symbols −∞ and +∞, and adopt the convention that −∞ < a < +∞ for all a ∈ R. In the sequel, the following notation is adopted. Notation 1. Let the symbols be as above. • For f ∈ K[x1 , . . . , xn ] and a semi-algebraic subset S of Rn , VR (f : S) stands for the subset {f (¯ α) | α ¯ ∈ S} of R. • For f ∈ K[x1 , . . . , xn ] and a finite subsets H of K[x1 , . . . , xn ], VR (f : H) stands for the subset {f (¯ α) | α ¯ ∈ Rn , and h(¯ α) = 0, ∀ h ∈ H} of R. • For f ∈ K[x1 , . . . , xn ] and two finite subsets G, H of K[x1 , . . . , xn ], VR (f : H/G) stands for the subset {f (¯ α) | α ¯ ∈ Rn , h(¯ α) = 0, ∀ h ∈ H, but g(¯ α) 6= 0, ∀ g ∈ G} of R. • For α, β ∈ R ∪ {−∞, +∞} with α < β, ]α, β[R stands for the open interval {γ ∈ R | α < γ < β} in R. [α, β]R stands for the closed interval {γ ∈ R | α 6 γ 6 β} in R. ]α, β]R stands for the half open-closed interval {γ ∈ R | α < γ 6 β} in R. [α, β[R stands for the half closed-open interval {γ ∈ R | α 6 γ < β} in R. • For a ¯ := (a1 , . . . , an ) ∈ Rn and a δ ∈ R with 0 < δ, OR (¯ α; δ) (or OR (a1 , . . . , an ; δ)) stands for the open box {(ξ1 , . . . , ξn ) ∈ Rn | −δ < ξi − ai < δ, i = 1, . . . , n}. For simplicity, the subscript R may be omitted in these symbols when R is just the field R of real numbers. By [2, Theorem 2.2.7], VR (f : S) is a semi-algebraic subset of R for f ∈ K[x1 , . . . , xn ] and a semialgebraic subset S of Rn . According to [2, Proposition 2.1.7], when S 6= ∅, VR (f : S) consists of finitely many disjoint (open, closed, half open-closed or half closed-open) intervals in R, if isolated points in VR (f : S) are regarded as closed intervals with the same endpoints. All the endpoints of these disjoint intervals are called the endpoints of VR (f : S). Naturally, The least endpoint of VR (f : S) is defined to be the infimum of VR (f : S) and is denoted by infVR (f : S). When VR (f : S) = ∅ (i.e., S = ∅), we adopt the convention that infVR (f : S) = +∞. Obviously, VR (f : H/G) is a semi-algebraic subset of R. So the symbol infVR (f : H/G) is well-defined. Likewise, infVR (f : H) is well-defined for f ∈ K[x1 , . . . , xn ] and a finite subset H of K[x1 , . . . , xn ]. According to the argument above, the equality-constrained minimization of polynomials over R is exactly to compute infVR (f : H) for f ∈ K[x1 , . . . , xn ] and a finite subset H of K[x1 , . . . , xn ], where K is a computable subfield of R. Obviously, infVR (f : H) is attained if and only if infVR (f : H) is a closed endpoint of VR (f : H). In what follows, we seek an effective method to find a finite subset U of univariate polynomials such that infVR (f : H) is a root of some polynomial in U in the case when infVR (f : H) is attained.

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Definition 3.1. Let f ∈ R[x1 , . . . , xn ] where R is a real closed field, and S a semi-algebraic subset n of R . A point a ¯ ∈ S is called an extremal point of f on S, if there exists a positive element δ in R such that either f (¯ z ) 6 f (¯ a) for all z¯ ∈ S ∩ OR (¯ a; δ) or f (¯ z ) > f (¯ a) for all z¯ ∈ S ∩ OR (¯ a; δ). In this case, f (¯ a) is called an extremal value of f on S. Notation 2. Let R be a real closed field, let K be a subfield of R, and f ∈ K[x1 , . . . , xn ]. • For a semi-algebraic subset S of Rn , EPointR (f : S) stands for the set of extremal points of f on S, and EValueR (f : S) stands for the set of extremal values of f on S. • For a finite subsets H of K[x1 , . . . , xn ], EPointR (f : H) stands for the set of extremal points of f on ZeroR (H), and EValueR (f : H) stands for the set of extremal values of f on ZeroR (H). • For two finite subsets G, H of K[x1 , . . . , xn ], EPointR (f : H/G) stands for the set of extremal points of f on ZeroR (H/G), and EValueR (f : H/G) stands for the set of extremal values of f on ZeroR (H/G). Likewise, the subscript R may be omitted in the above symbols when R is just the field R of real numbers. Without difficulty, we may establish the following proposition, whose proof is trivial. Proposition 3.2. Let R be a real closed field, let K be a subfield of R, and f ∈ K[x1 , . . . , xn ]. Then the following statements are true: (1) If H is a finite subsets of K[x1 , . . . , xn ] such that inf VR (f : H) is attained, then inf VR (f : H) ∈ EValueR (f : H). (2) If S and T are two semi-algebraic subsets of Rn such that T ⊆ S, and a ¯ ∈ EPointR (f : S), then a ¯ ∈ EPointR (f : T ) whenever a ¯ ∈ T. (3) If S and T are two semi-algebraic subsets of Rn and T is open, then EPointR (f : S ∩ T ) ⊆ EPointR (f : S). (4) If G, H are two finite subsets of K[x1 , . . . , xn ], then EPointR (f : H/G) ⊆ EPointR (f : H). Now let H := {h1 , . . . , hs } be a finite subsets of K[x1 , . . . , xn ]. Then we obtain the s × n matrix as follows:  ∂h  ∂h1 ∂h1 1 · · · ∂x ∂x2 ∂xn  .1 .. ..  ..  .. Jacob(H) :=  . . . .  ∂hs ∂h1 ∂hs · · · ∂x ∂x1 ∂x2 n As usual, the matrix Jacob(H) is called the Jacobian matrix of H (with respect to the variables x1 , . . . , xn ). For a point a ¯ := (a1 , . . . , an ) ∈ Rn , denote by Jacob(H)(¯ a) the matrix obtained from Jacob(H) by substituting xi = ai for i = 1, . . . , n. By the familiar Transfer principle for real closed fields (see [1, Theorem 2.78] or [2, Proposition 5.2.3]), we can establish the following theorem. Theorem 3.3 (Lagrange’s theorem for real closed fields). Let R be a real closed field, let f ∈ R[x1 , . . . , xn ], H := {h1 , . . . , hs } a finite subset of R[x1 , . . . , xn ]. If a ¯ ∈ EPointR (f : H) and the matrix Jacob(H)(¯ a) has rank s, then there exist λ1 , . . . , λs in R such that ∂h1 ∂hs ∂f (¯ a) − λ1 (¯ a) − · · · − λs (¯ a) = 0, ∂xi ∂xi ∂xi

i = 1, . . . , n.

Proof. Since the matrix Jacob(H)(¯ a) has rank s, we have s 6 n. Let c1 , . . . , cm be all the coefficients of f, h1 , . . . , hs , and introduce the corresponding variables y1 , . . . , ym . ˆ 1 , . . . , ˆhs in Q[x1 , . . . , xn , y1 , . . . , ym ] Replacing ci with yi for i = 1, . . . , m, we can obtain the polynomials fˆ, h ˆ ˆ such that f (x1 , . . . , xn , c1 , . . . , cm ) = f and hk (x1 , . . . , xn , c1 , . . . , cm ) = hk for k = 1, . . . , s. Then the ˆ 1, . . . , h ˆ s with respect to variables x1 , . . . , xn as follows: Jacobian matrix of h  ∂ hˆ ˆ1 ˆ1  ∂h ∂h 1 · · · ∂x ∂x1 ∂x2 n  .. ..  .. b :=  ..  Jacob(H) . . . .  . ˆs ˆ1 ˆs ∂h ∂h ∂h · · · ∂x ∂x1 ∂x2 n

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b where r := Denote by ∆1 , . . . , ∆r all the s×s minors of Jacob(H),

∂ fˆ ∂xi

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, and write Ψi for the polynomial

n s

h1 − z1 ∂∂x − · · · − zs ∂∂xhsi in Q[x1 , · · · xn , y1 , . . . , ym , z1 , . . . , zs ] for i = 1, . . . , n. i Since R is a real closed, the characteristic of R is 0. Hence we may assume Q ⊆ R. Write R0 and R0′ for the algebraic closures of Q in R and R, respectively. By [18, Lemma 3.13], both R0 and R0′ are the real closures of the ordered field (Q, 6), where 6 is the ordinal ordering of Q. By the uniqueness of real closures of an ordered field (see [18, Theorem 3.10]), R0′ is order-preserving isomorphic to R0 . So we may assume R0′ ≡ R0 . Now consider the sentence in the language of ordered fields with constants in Q(⊆ R0 ) as follows:  _  ^  ∀ (x1 , . . . xn , y1 , . . . , ym ) ∆k 6= 0 → ∃(z1 , . . . , zs ) Ψi = 0 . 16k6r

16i6n

By the classical method of Lagrange multipliers (see [22, Theorem 5.1]), the above sentence is true in R. By the Transfer principle for real closed fields (see [1, Theorem 2.78] or [2, Proposition 5.2.3]), this sentence also is true in R0 . Again by the Transfer principle, this sentence is true in R, i.e., Theorem 3.3 is true. This completes the proof. Now let C = [g1 , . . . , gs ] be an incomplete chain in F [x1 , . . . , xn ] (with respect to the usual lexicographic order). Then the subset ℧ of {1, . . . , s} is nonempty, where ℧ := {k | 1 6 k 6 s, and xk ≺ mv(gk )}. Denote by µ the least integer in ℧. Obviously, mv(gk ) = xk ,

k = 1, . . . , µ − 1, and xµ ≺ mv(gµ ) ≺ · · · ≺ mv(gs ).

Rearrange the variables xµ , xµ+1 , . . . , xn as follows: xj1 , . . . , xjs−µ+1 , xjs−µ+2 , . . . , xjn−µ+1 , where xjk := mv(gµ+k−1 ) for k = 1, . . . , s − µ + 1, and xjs−µ+2 , . . . , xjn−µ+1 are the other variables. For f ∈ K[x1 , . . . , xn ], consider the (s − µ + 2) × (s − µ + 1) matrix as follows:   ∂f ∂f ∂f · · · ∂xjs−µ+1   ∂xj1 ∂xj2 ∂gµ ∂gµ   ∂gµ · · ·  ∂xj1 ∂xj2 ∂xjs−µ+1   . M(f,C) :=  .  .. .. ..  ..  . . .   ∂gs ∂xj1

∂gs ∂xj2

···

∂gs ∂xjs−µ+1

In what follows, write ∆i for the determinant of the (s − µ + 1) × (s − µ + 1) matrix obtained by Q ∂gµ+k−1 deleting the (i + 1)-th row in M(f,c) , i = 1, . . . , s − µ + 1, and put ∆C := 16k6s−µ+1 ∂x . For jk i ∈ {s − µ + 2, . . . , n − µ + 1}, we can obtain the polynomial in F [x1 , . . . , xn ] as follows: ∆C ·

s−µ+1 X ∂gjµ+k−1 ∂f + (−1)k ∆k · . ∂xji ∂xji k=1

Definition 3.4. Let f , C and the relevant symbols be as above. The set of Lagrangian polynomials of f and C is defined as follows:  s−µ+1 X ∂gjµ+k−1 ∂f ∆C · + (−1)k ∆k · ∂xji ∂xji k=1

 i = s − µ + 2, . . . , n − µ + 1 .

When C = [g1 , . . . , gs ] is a complete chain in F [x1 , . . . , xn ], the set of Lagrangian polynomials of f and C is defined to be the empty set { }. In what follows, denote by L(f, C) the set of the Lagrangian polynomials of f and C.

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The following proposition can be found as the corollary of [32, Proposition 1]. Proposition 3.5. Let f (x1 , x2 , . . . , xn ) ∈ F [x1 , x2 , . . . , xn ] where F is a field of characteristic 0, let z ∂f , i = 1, . . . , n. Then be a new variable, and J the ideal of F [z, x1 , x2 , . . . , xn ] generated by z − f and ∂x i J ∩ F [z] 6= {0}. For the purpose of this paper, we establish such a generalization of Proposition 3.5 as follows. Proposition 3.6. Let f (x1 , x2 , . . . , xn ) ∈ F [x1 , x2 , . . . , xn ], where F is a field of characteristic 0, let C := [g1 , . . . , gm ] be a complete chain in F [x1 , . . . , xm ], where 0 < m 6 n, and z a new variable. ∂f If J is the ideal of F [z, x1 , x2 , . . . , xn ] generated by z − f, g1 , . . . , gm and ∂x , i = m + 1, . . . , n, then i (J : Π) ∩ F [z] 6= {0}, where Π is the product of the initials of members in C. √ Proof. Since F [z, x1 , x2 , . . . , xn ] is a Noetherian ring, the radical J of J can be represented in the form \ √ J= ℘k , 16k6s

where ℘k is a prime ideal of F [z, x1 , x2 , . . . , xn ], k = 1, . . . , s. √ When Π ∈ J, it √ is obvious that 1 ∈ J : Π. So we have 1 ∈ (J : Π) ∩ F [z], and (J : Π) ∩ F [z] 6= {0}. Now assume Π ∈ / J. Without loss of generality, we may assume that Π ∈ / ℘k for k = 1, . . . , r but Π ∈ ℘k for k = r + 1, . . . , s, where 1 6 r 6 s. It follows that ℘k : Π = ℘k for k = 1, . . . , r but ℘k : Π = F [x1 , x2 , . . . , xn , z] for k = r + 1, . . . , s. So we have  \  \ \ √ J :Π= ℘k : Π = (℘k : Π) = ℘k . 16k6s

16k6s

16k6r

Let ℘ be any prime ideal among ℘1 , . . . , ℘r , and put q := ℘ ∩ F [z, x1 , x2 , . . . , xm ]. Obviously, q is a prime ideal of F [z, x1 , x2 , . . . , xm ] such that Π ∈ / q. Denote by E the fraction field of the domain F [z, x1 , . . . , xm ]/q, put α := z + q, and put βi := xi + q for i = 1, . . . , m. Write ℓi (x1 , . . . , xi−1 ) for the initial of gi for i = 1, . . . , m. Then ℓi (x1 , . . . , xi−1 ) ∈ / q, and ℓi (β1 , . . . , βi−1 ) 6= 0, i = 1, . . . , m. Observe that gi ∈ q for i = 1, . . . , m. It follows that gi (β1 , . . . , βi−1 , βi ) = 0, i = 1, . . . , m. This implies that all the elements β1 , . . . , βm are algebraic over F . Hence F [β1 , . . . , βm ] = F (β1 , . . . , βm ), i.e., the ring F [β1 , . . . , βm ] is actually a field. Put f ∗ := f (β1 , . . . , βm , xm+1 , . . . , xn ), and let J ∗ be the ideal of F [β1 , . . . , βm ][z, xm+1 , . . . , xn ] gener∗ ∗ ated by z − f ∗ and ∂f ∂xi , i = m+ 1, . . . , n. By Proposition 3.5, J ∩F [β1 , . . . , βm ][z] 6= {0}. In other words, ∗ there is a non-zero polynomial φ(z, β1 , . . . , βm ) in J ∩ F [β1 , . . . , βm ][z], where φ ∈ F [z, x1 , . . . , xm ]. By the definition of J ∗ , we have φ(z, β1 , . . . , βm ) = u0 (z, β1 , . . . , βm , xm+1 , . . . , xn )(z − f ∗ ) n X ∂f ∗ + ui (z, β1 , . . . , βm , xm+1 , . . . , xn ) , ∂xi i=m+1 where u0 , ui ∈ F [z, x1 , . . . , xn ] for i = m + 1, . . . , n. Observe that q = ℘ ∩ F [z, x1 , x2 , . . . , xm ]. By the second isomorphism theorem for rings (see [9, p. 108]), there exists an isomorphism of F [z, x1 , x2 , . . . , xm ]/q onto (F [z, x1 , x2 , . . . , xm ] + ℘)/℘ such that w + q 7→ w + ℘ for all w ∈ F [z, x1 , x2 , . . . , xm ]. Denote by L the fraction field of the domain F [z, x1 , . . . , xn ]/℘. Then (F [z, x1 , x2 , . . . , xm ] + ℘)/℘ ⊆ F [z, x1 , x2 , . . . , xn ]/℘ ⊆ L. By identifying F [z, x1 , x2 , . . . , xm ]/q with (F [z, x1 , x2 , . . . , xm ] + ℘)/℘, F [z, x1 , x2 , . . . , xm ]/q may be regarded as a subring of L, and E is hence a subfield of L. In this case, α = z + q ≡ z + ℘ and βi = xi + q ≡ xi + ℘ for i = 1, . . . , m. Put βi := xi + ℘ for i = m + 1, . . . , n. Then α, βi ∈ L for i = 1, . . . , n. Using the substitution that z = α and xi = βi for i = m + 1, . . . , n in the preceding equality, we have

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φ(α, β1 , . . . , βm ) = u0 (α, β1 , . . . , βn )(α − f (β1 , . . . , βn )) n X ∂f (β1 , . . . , βn ). + ui (α, β1 , . . . , βn ) ∂xi i=m+1 ∂f ∂f Since z − f , ∂x∂f , . . . , ∂x ∈ J ⊆ ℘, we have α − f (β1 , . . . , βn ) = 0 and ∂x (β1 , . . . , βn ) = 0 for m+1 n i i = m + 1, . . . , n. So we get φ(α, β1 , . . . , βm ) = 0. Since φ(z, β1 , . . . , βm ) is a non-zero polynomial over F (β1 , . . . , βm ) in one variable z, α is algebraic over F (β1 , . . . , βm ). Since β1 , . . . , βm are algebraic over F , α is algebraic over F . Denote by h0 (z) the minimal polynomial of α over the field F . Then h0 (z) ∈ F [z] and h0 (α) = 0. Observe that h0 (z) + ℘ = h0 (z + ℘) = h0 (α) = 0 ≡ 0 + ℘. It follows that h0 (z) ∈ ℘, and h0 (z) ∈ ℘ ∩ F [z]. By the arbitrariness of ℘, there exists a non-zero polynomial hk (z) such √ that T hk (z) ∈ ℘k ∩ F [z], k = 1, . . . , r. Put h(z) := h1 (z) · · · hr (z). Then h(z) ∈ 16k6r ℘k , i.e., h(z) ∈ J : Π. √ So we have Πλ h(z) ∈ J for some positive integer λ. Hence (Πλ h(z))d ∈ J for some positive integer d, and hd (z) ∈ J : Π. Thus hd (z) ∈ (J : Π) ∩ F [z]. This completes the proof.

Theorem 3.7. Let R be a real closed field with subfield K, let f ∈ K[x1 , . . . , xn ], let C = [g1 , . . . , gs ] be a chain in K[x1 , . . . , xn ], and a ¯ ∈ EPointR (f : C/Q) where Q is the initial-separant set of C. Then the following statements are true: (1) If C is complete, i.e., s = n, then f (¯ a) is a root of u(t) where u(t) is the greatest common divisor of all coefficients of RvRes(t − f, C) as a polynomial over K[t] in variables x1 , . . . , xn . (2) If C is incomplete and D1 , . . . , Dr is a triangular decomposition of C ∪ L(f, C), then f (¯ a) is a root of some polynomial among u1 (t), . . . , ur (t), where uk (t) is the greatest common divisor of all coefficients of RvRes(t − f, Dk ) as a polynomial over K[t] in variables x1 , . . . , xn for k = 1, . . . , r. Proof. Write Π for the product of the initials of members in C, and ΠQ for the product of all the members in Q. Obviously, Π is a divisor of ΠQ . By the definition of EPointR (f : C/Q), we have a ¯ ∈ EPointR (f : C/Q) ⊆ ZeroR (C/Q) = ZeroR (C/{ΠQ }) ⊆ ZeroR (C/{Π}). (1) Let J be the ideal of K[t, x1 , x2 , . . . , xn ] generated by t − f, g1 , . . . , gn . By Proposition 3.6, we have (J : Π) ∩ K[z] 6= {0}. Observe that C = {g1 , . . . , gn } is already a chain and a ¯ ∈ ZeroR (C/Q) ⊆ ZeroR (C/I), where I is the initial set of C. By Theorem 2.10 and its proof, Statement (1) is true. (2) By the preceding argument, there is a positive integer µ in {1, . . . , s} such that mv(gk ) = xk , k = 1, . . . , µ − 1, and xk ≺ mv(gk )

for k = µ, . . . , s.

As before, rearrange the variables xµ , xµ+1 , . . . , xn as follows: xj1 , . . . , xjs−µ+1 , xjs−µ , . . . , xjn−µ+1 , where xjk := mv(gµ+k−1 ) for k = 1, . . . , s − µ + 1, and xjs−µ , . . . , xjn−µ+1 are the other variables. Observe that the partial derivative Hence

∂gµ+k−1 ∂xjk

∂gµ+k−1 ∂xjk

is just the separant of gµ+k−1 for k = 1, . . . , s − µ + 1.

∈ Q, k = 1, . . . , s − µ + 1. By the hypothesis that a ¯ ∈ EPointR (f : C/Q), we have ∂gµ+k−1 a) ∂xjk (¯

6= 0, k = 1, . . . , s − µ + 1.

Put a ¯ := (a1 , . . . , an ), and write h|a¯ for the polynomial h(a1 , . . . , aµ−1 , xµ , . . . , xn ) in R[xµ , . . . , xn ] for h ∈ K[x1 , . . . , xn ]. By the hypothesis, a ¯′ is obviously an extremal point of f |a¯ on ZeroR (C ′ /I ′ ) where ′ ′ a ¯ := (aµ , . . . , an ), C := [gµ |a¯ , . . . , gn |a¯ ] and I ′ is the initial set of C ′ as a chain in R[xµ , . . . , xn ]. By Proposition 3.2, a ¯′ is an extremal point of f |a¯ on ZeroR (C ′ ). Observe that the Jacobian matrix of C ′ (with respect to the variables xj1 , . . . , xjn−µ+1 ) is the (s − µ + 1) × (n − µ + 1) matrix as follows:  ∂g |  ∂gµ |a ∂g | µ a ¯ ¯ · · · ∂xj µ a¯ ∂xj1 ∂xj2 n−µ+1    .  .. .. .. Jacob(C ′ ) :=  .. . . . .   ∂gs |a ∂g | ∂g | ¯ s a ¯ · · · ∂xj s a¯ ∂xj ∂xj 1

2

n−µ+1

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It is easy to see that the first (s − µ + 1) columns in Jacob(C ′ ) constitute a lower triangular (s − Q ∂gµ+k−1 . Observe that µ + 1) × (s − µ + 1) matrix with determinant ∆C |a¯ , where ∆C := 16k6s−µ+1 ∂x jk Q ∂gµ+k−1 ′ ′ ′ ∆C |a¯ (¯ a ) = 16k6s−µ+1 ∂xj (¯ a) 6= 0. Hence, the rank of Jacob(C )(¯ a ) is equal to s − µ + 1. By k Lagrange’s theorem for real closed fields (see Theorem 3.3), there exist λ1 , . . . , λs−µ+1 in R such that ∂f |a¯ ′ ∂gµ |a¯ ′ ∂gs |a¯ ′ (¯ a ) − λ1 (¯ a ) − · · · − λs−µ+1 (¯ a ) = 0, ∂xji ∂xji ∂xji

i = 1, . . . , n − µ + 1.

It follows that λ1

∂gµ ∂gs ∂f (¯ a) + · · · + λs−µ+1 (¯ a) = (¯ a), ∂xji ∂xji ∂xji

i = 1, . . . , n − µ + 1.

(3.1)

This implies that (λ1 , . . . , λs−µ+1 ) is a solution of the system of linear equations as follows: y1

∂gµ ∂gs ∂f (¯ a) + · · · + ys−µ+1 (¯ a) = (¯ a), ∂xji ∂xji ∂xji

i = 1, . . . , s − µ + 1.

Obviously, the coefficient matrix of this system is an upper triangular matrix with determinant ∆C (¯ a). Observe that ∆C (¯ a) 6= 0. By the familiar Cramer’s rule, we have λi = (−1)i+1

∆i (¯ a) , i = 1, . . . , s − µ + 1, ∆C (¯ a)

where, for i = 1, . . . , s − µ + 1, ∆i is the determinant of the (s − µ + 1) × (s − µ + 1) matrix obtained by deleting the (i + 1)-th row in   ∂f ∂f ∂f · · · ∂xjs−µ+1  ∂xj1 ∂xj2  ∂gµ ∂gµ  ∂gµ  · · ·  ∂xj1 ∂xj2 ∂xjs−µ+1   . M(f,C) :=  .  .. .. ..  ..  . . .   ∂gs ∂xj1

∂gs ∂xj2

···

∂gs ∂xjs−µ+1

∂g

∂f ∂gs Put Li := ∆C ∂x − ∆1 ∂xjµ + · · · + (−1)s−µ+1 ∆s−µ+1 ∂x for i = s − µ + 2, . . . , n − µ + 1. According ji ji i to Definition 3.4, L(f, C) = {Ls−µ+2 , . . . , Ln−µ+1 }. By the equalities in (3.1), we can get

  ∂f ∂gµ ∂gs ∆C − ∆1 + · · · + (−1)s−µ+1 ∆s−µ+1 (¯ a) ∂xji ∂xji ∂xji   ∂f ∂gµ ∂gs = ∆C (¯ a) (¯ a) − λ1 (¯ a) − · · · − λs−µ+1 (¯ a) = 0, ∂xji ∂xji ∂xji

Li (¯ a) =

where i = s − µ + 2, . . . , n − µ + 1. So we have a ¯ ∈ ZeroR (C ∪ L(f, C)/{ΠQ }). Denote by Φ the polynomial in K[y1 , . . . , ys−µ+1 , x1 , . . . , xn ] as follows: Φ := f −

s−µ+1 X

yk · gµ+k−1 .

k=1

Let J1 be the ideal of K[t, y1 , . . . , ys−µ+1 , x1 , . . . , xn ] generated by    ∂Φ ∂Φ {t − Φ} ∪ {g1 , . . . , gµ−1 } ∪ k = 1, . . . , s − µ + 1 ∪ ∂yk ∂xjk

 k = 1, . . . , n − µ + 1 .

By Proposition 3.6, we have (J1 : Π0 ) ∩ K[t] 6= {0}, where Π0 is the product of the initials of g1 , . . . , gµ−1 . Obviously,   ∂Φ {g1 , . . . , gµ−1 } ∪ k = 1, . . . , s − µ + 1 = {g1 , . . . , gs }. ∂yk

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It is easy to see that J1 is actually the ideal of K[t, y1 , . . . , ys−µ+1 , x1 , . . . , xn ] generated by   ∂Φ k = 1, . . . , n − µ + 1 . {t − f } ∪ {g1 , . . . , gs } ∪ ∂xjk Hence, there exists a non-zero polynomial h(t) in K[t] such that Πℓ0 h(t) = w0 · (t − f ) +

s X

wi · gi +

i=1

n−µ+1 X

vk ·

k=1

∂Φ , ∂xjk

(3.2)

where ℓ > 0, and w0 , w1 , . . . , ws , v1 , . . . , vn−µ+1 ∈ K[t, y1 , . . . , ys−µ+1 , x1 , . . . , xn ]. Now consider the following system of linear equations over the field K(x1 , . . . , xn ) of rational functions in variables y1 , . . . , ys−µ+1 : ∂Φ = 0, k = 1, . . . , s − µ + 1, ∂xjk i.e., s−µ+1 X k=1

∂f gµ+k−1 · yk = , ∂xjk ∂xjk

k = 1, . . . , s − µ + 1.

By Cramer’s rule, it is similar to prove that the system has its solution as follows: ∆i yi = (−1)i+1 , i = 1, . . . , s − µ + 1, ∆C where ∆C and ∆i ’s are as above. ∆i (i = 1, . . . , s − µ + 1) in (3.2), we can get Substituting yi with (−1)i+1 ∆ C Πℓ0 h(t) = w0∗ · (t − f ) +

s X i=1

wi∗ · gi +

n−µ+1 X

k=s−µ+2

vk∗ ·

Lk , ∆C

(3.3)

∆i where ψ ∗ stands for the expression obtained by substituting yi with (−1)i+1 ∆ (i = 1, . . . , s − µ + 1) for C ψ ∈ K[t, y1 , . . . , ys−µ+1 , x1 , . . . , xn ]. ∗ Choose a sufficiently large positive integer m such that ∆m C wi ∈ K[t, x1 , . . . , xn ] for i = 0, 1, . . . , s m−1 ∗ and ∆C vk ∈ K[t, x1 , . . . , xn ] for k = s − µ + 2, . . . , n − µ + 1. By (3.3), it is easy to see that ℓ ∆m C Π0 h(t) ∈ J, where J is the ideal of K[t, x1 , . . . , xn ] generated by {t − f } ∪ C ∪ L(f, C). Hence h(t) ∈ J : ∆C Π0 , and (J : ∆C Π0 ) ∩ K[t] 6= {0}. Since both ∆C and Π0 are divisors of ΠQ , it is easy to see that a ¯ ∈ ZeroR (C ∪ L(f, C)/{ΠQ }) ⊆ ZeroR (C ∪ L(f, C)/{∆C Π0 }). By Theorem 2.10, Statement (2) is true. This completes the proof.

According to Definition 2.11, Theorem 3.7 can be described in the following version. Theorem 3.8. Let R be a real closed field with subfield K, let f ∈ K[x1 , . . . , xn ], let C = [g1 , . . . , gs ] be a chain in K[x1 , . . . , xn ], and let a ¯ ∈ EPointR (f : C/Q), where Q is the initial-separant set of C. Then the following statements are true: (1) If C is complete, then f (¯ a) is a root of u(t), where {u(t)} is the output of the algorithm CUP adapted to (f, C). (2) If C is incomplete and D1 , . . . , Dr is a triangular decomposition of C ∪ L(f, C), then f (¯ a) is a root of some polynomial among u1 (t), . . . , ur (t), where {uk (t)} is the output of the algorithm CUP adapted to (f, Dk ), k = 1, . . . , r. Now, we are able of establishing the desired algorithm as follows: Algorithm 2 (Catching the constrained minimums of polynomials). Structure: A computable ordered field (K, 6) with real closed extension R. Input: A polynomial f ∈ K[x1 , . . . , xn ] and a finite subset H of K[x1 , . . . , xn ]. Output: A finite set U of univariate polynomials in K[t] such that • inf V(f : H) is a root of some polynomial in U whenever inf V(f : H) is attained.

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Procedure: Step 1. By the Wu’s algorithm for Zero-Decomposition Theorem II, compute a triangular decomposition C1 , . . . , Cr of H with seperants. Step 2. For i from 1 to r, create a finite subset Ui of K[t] in the following manner: If Ci is complete, compute Ui as the output of the algorithm CUP adapted to (f, Ci ) according to Definition 2.11. Else, implement the following computations: (1) Compute a triangular decomposition D1 , . . . , Dsi of Ci ∪ L(f, Ci ); (2) Compute the output Uik of the algorithm CUP adapted to (f, Dk ) for k = 1, . . . , si , and set S Ui := 16k6si Uik . S Step 3. RETURN (U := 16i6r Ui ).

Proof of correctness. Assume that inf V(f : H) is attained. Then inf V(f : H) = f (¯ α) for some S a ¯ ∈ ZeroR (H). By the computation in Step 1, ZeroR (H) = 16i6r ZeroR (Ci /Qi ), where Qi is the initial-separant set Ci for i = 1, . . . , r. It follows that a ¯ ∈ ZeroR (Cj /Qj ) for some j ∈ {1, . . . , r}. By Proposition 3.2(1), α ¯ ∈ EPointR (f : H). By Proposition 3.2(2), we further have α ¯ ∈ EPointR (f : Cj /Qj ). Let Uj is the result computed in Step 2. By Theorem 3.8, inf V(f : H) is a root of some polynomial in Uj . By Step 3, we have Uj ⊆ U . This completes the proof.

Remark 3.9. By Algorithm 2 and its proof of correctness, it is easy to see that every extremal value in EValueR (f : H) is a root of some polynomial in U , if U is the output of Algorithm 2 for an input (f, H). In particular, every extremal value in EValueR (f : H) is algebraic over the field K whenever all the coefficients of f and the polynomials in H are contained in K. 3.2

The general case

In this subsection, we consider the general case when there is no assumption that inf V(f : H) is attained. As a toy example in the case when inf V(f : H) is not attained, we put f := (xy − 1)2 + x2 + z and H := {z}. Since limx→0 f (x, x1 , 0) = limx→0 x2 = 0, it is easy to see inf V(f : H) = 0. However, it is obvious that inf V(f : H) is not attained. Firstly, we extend the field R of real numbers to a real closed field containing infinitely large elements and infinitesimal elements. Let η1 , . . . , ηn be n indeterminates over R, and write R(η1 , . . . , ηi ) for the fraction field of the polynomial ring R[η1 , . . . , ηi ] for i = 1, . . . , n. Then the ordering 6 of R may be extended to such an ordering of R(η1 , . . . , ηn ), still denoted by 6, in the following manner: For non-zero g, h ∈ R[η1 , . . . , ηn ], hg < 0, if and only if lc(gh; η1 , . . . , ηn ) < 0, where lc(u; η1 , . . . , ηn ) stands for the leading coefficient of u with respect to the lexicographic order η1 ≺ · · · ≺ ηn for any non-zero u ∈ R[η1 , . . . , ηn ]. It is easy to see that ηi is a positive and infinitely large element over the subfield R(η1 , . . . , ηi−1 ) in the sense that w < ηi for all w ∈ R(η1 , . . . , ηi−1 ), i = 1, . . . , n. According to [18, Theorem 3.10], denote by R the real closure of the ordered field (R(η1 , . . . , ηn ), 6), and the unique ordering of R is still denoted by 6. Now, we construct the two subsets of R as follows: A := {α ∈ R | For some positive number d ∈ R, −d 6 α 6 d}, M := {α ∈ R | For every positive number d ∈ R, −d 6 α 6 d}. Hence, M consists of all elements in R which are “infinitesimal” over R. Obviously, R ⊂ A. By the definition of 6, we further have ηi−1 ∈ M for i = 1, . . . , n. Moreover, it is easy to verify that A is a subring of R and M is an ideal of A. For α ∈ R, we obtain the subset of R as follows: Ωα := {r ∈ R | α 6 r}. Write inf(Ωα ) for the infimum of Ωα . Then inf(Ωα ) ∈ R ∪ {+∞, −∞}, and inf(Ωα ) ∈ R if and only if α ∈ A. So there is a mapping π of R into R ∪ {+∞, −∞} such that π(α) = inf(Ωα ) for all α ∈ R. It is easy to verify that the following proposition is true.

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Proposition 3.10. Let the symbols be as above. Then the following conditions are satisfied: (1) The restricted mapping π |A is an R-homomorphism of the ring A onto R, and M is exactly the kernel of π |A . (2) For ξ ∈ R \ A, π(ξ) = +∞ if ξ > 0; otherwise, π(ξ) = −∞. (3) For α, β ∈ R, α 6 β implies π(α) 6 π(β), where we adopt the convention that −∞ < r < +∞ for all r ∈ R. Q In the affine space Rn , denote by ni=1 [−ηi , ηi ]R the closed cuboid as follows: {(a1 , . . . , an ) ∈ Rn | −ηi 6 ai 6 ηi , i = 1, . . . , n}.

Qn Then the interior of i=1 [−ηi , ηi ]R is the open cuboid as follows: Qn n i=1 ] − ηi , ηi [R := {(a1 , . . . , an ) ∈ R | −ηi < ai < ηi , i = 1, . . . , n}. Qn Obviously, i=1 [−ηi , ηi ]R is a closed and bounded semi-algebraic subset of Rn such that Rn ⊆ Qn i=1 [−ηi , ηi ]R .

Proposition 3.11. Let f ∈ R[x1 , . . . , xn ], and H a finite subset of R[x1 , . . . , xn ]. If ZeroR (H) 6= ∅, then Qn Qn inf R (f : i=1 [−ηi , ηi ]R ∩ ZeroR (H)) is attained, i.e., f has its minimum on i=1 [−ηi , ηi ]R ∩ ZeroR (H). Q Proof. Let φ be the mapping of ni=1 [−ηi , ηi ]R ∩ ZeroR (H) into R such that (α1 , . . . , αn ) 7→ f (α1 , . . . , αn ) for

(α1 , . . . , αn ) ∈

n Y

[−ηi , ηi ]R ∩ ZeroR (H).

i=1

Obviously, the mapping φ is continuous for the interval topology of R. By [2, Theorem 2.5.8], Y  n φ [−ηi , ηi ]R ∩ ZeroR (H) i=1

is a closed and bounded semi-algebraic subset of R. According to [2, Proposition 2.1.7], Y  n φ [−ηi , ηi ]R ∩ ZeroR (H) i=1

Q consists of a finite number of disjoint closed intervals, whenever the isolated points in φ( ni=1 [−ηi , ηi ]R ∩ZeroR (H)) are regarded as closed intervals with same endpoint. It is clear that the least element among Q all the endpoints of these closed intervals is just the minimum of f on ni=1 [−ηi , ηi ]R ∩ ZeroR (H). This completes the proof. Remark 3.12. Let H be a finite subset of R[x1 , . . . , xn ] such that ZeroR (H) = ∅. By the Transfer Qn principle for real closed fields, we have ZeroR (H) = ∅, and i=1 [−ηi , ηi ]R ∩ ZeroR (H) = ∅. So we may Qn adopt the convention that f has its minimum +∞ on i=1 [−ηi , ηi ]R ∩ ZeroR (H) for f ∈ R[x1 , . . . , xn ]. Thus the assumption that ZeroR (H) 6= ∅ is superfluous in Proposition 3.11. Proposition 3.13. Let f ∈ R[x1 , . . . , xn ], and let H be a finite subset of R[x1 , . . . , xn ] such that Qn ZeroR (H) 6= ∅. If µ is the minimum of f on i=1 [−ηi , ηi ]R ∩ ZeroR (H), then following statements are true: (1) π(µ) = inf V(f : H), i.e., π(µ) is the constrained infimum of f on ZeroR (H), where π is the mapping as defined as above. (2) inf V(f : H) is attained if and only if µ ∈ R. In this case, µ is the constrained minimum of f on ZeroR (H). Q Proof. Let (α1 , . . . , αn ) be a minimum point of f on ni=1 [−ηi , ηi ]R ∩ ZeroR (H). Then −ηi 6 αi 6 ηi for i = 1, . . . , n, and f (α1 , . . . , αn ) = µ. (1) Since ZeroR (H) 6= ∅, we have inf V(f : H) ∈ R, or inf V(f : H) = −∞. Consider the following possible cases:

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Case 1. inf V(f : H) ∈ R. Put H := {h1 , . . . , hs }, and put a := inf V(f : H). By the definition of infimums, the sentence is true in R:  ^  ∀ (x1 , . . . , xn ) hk (x1 , . . . , xn ) = 0 → f (x1 , . . . , xn ) > a . 16k6s

Observe that R ⊆ R. By the Transfer principle for real closed fields, the above sentence also is true in R. It follows that µ = f (α1 , . . . , αn ) > a, and µ − a > 0. Let δ be an arbitrary positive number. According to the definition of infimums, f (a1 , . . . , an ) < a + δ Q for some (a1 , . . . , an ) ∈ ZeroR (H). Observe that (a1 , . . . , an ) ∈ ZeroR (H) ⊆ ni=1 [−ηi , ηi ]R ∩ ZeroR (H). Qn It follows that µ 6 f (a1 , . . . , an ), because µ is the minimum of f on i=1 [−ηi , ηi ]R ∩ ZeroR (H). Hence µ < a + δ, and µ − a < δ. It follows that 0 6 µ − a < δ. By the arbitrariness of δ, we have µ − a ∈ M. By Proposition 3.10(1), we have π(µ − a) = 0, and π(µ) − a = 0. Thus π(µ) = a = inf V(f : H). Case 2. inf V(f : H) = −∞. Suppose that π(µ) 6= −∞. By Proposition 3.10(2), either µ > 0 or µ ∈ A. So we always have µ > d for some d ∈ R. Let (z1 , . . . , zn ) be an arbitrary zero in ZeroR (H). Then Qn Qn (z1 , . . . , zn ) ∈ i=1 [−ηi , ηi ]R ∩ ZeroR (H). Since µ is the minimum of f on i=1 [−ηi , ηi ]R ∩ ZeroR (H), we have f (z1 , . . . , zn ) > µ > d. From the definition of inf V(f : H), it follows that inf V(f : H) > d, a contradiction. Thus π(µ) = −∞ = inf V(f : H). (2) Assume that f has its minimum b on ZeroR (H). Then there exists a zero (a1 , . . . , an ) in ZeroR (H) Qn such that f (a1 , . . . , an ) = b. Observe that (a1 , . . . , an ) ∈ ZeroR (H) ⊆ i=1 [−ηi , ηi ]R ∩ ZeroR (H). It Q follows that µ 6 b, because µ is the minimum of f on ni=1 [−ηi , ηi ]R ∩ ZeroR (H). Moreover, since b is the minimum of f on ZeroR (H), the following sentence is true in R:  ^  ∀ (x1 , . . . , xn ) hk (x1 , . . . , xn ) = 0 → f (x1 , . . . , xn ) > b . 16k6s

Observe that R ⊆ R. By the Transfer principle for real closed fields, the above sentence also is true in R. So µ = f (α1 , . . . , αn ) > b. Thus µ = b ∈ R. Conversely, assume that µ ∈ R. Since f (α1 , . . . , αn ) = µ, the following sentence is true in R:  ^   ∃ (x1 , . . . , xn ) hk (x1 , . . . , xn ) = 0 ∧ f (x1 , . . . , xn ) = µ . 16k6s

Observe that the constant µ in this sentence is a real number. By the Transfer principle for real closed fields, this sentence also is true in R. Hence, there exists a (b1 , . . . , bn ) ∈ Rn such that f (b1 , . . . , bn ) = µ. Qn For any z1 , . . . , zn ∈ ZeroR (H), (z1 , . . . , zn ) ∈ i=1 [−ηi , ηi ]R ∩ ZeroR (H). Since µ is the minimum Qn of f on i=1 [−ηi , ηi ]R ∩ ZeroR (H), we have f (z1 , . . . , zn ) > µ. This implies that µ is the constrained minimum of f on ZeroR (H). This completes the proof. For the sake of convenience, we need the following definition. Definition 3.14. Let m be a positive integer with 1 6 m < n. A sequence [j1 , . . . , jm ] of positive integers is called a subsequence of [1, . . . , n] if 1 6 j1 < · · · < jm 6 n. The sequence [jm+1 , . . . , jn ] is called the complement of [j1 , . . . , jm ] in [1, . . . , n] if {1, . . . , n} = {j1 , . . . , jm , jm+1 , . . . , jn } and jm+1 < · · · < jn . Let [j1 , . . . , jm ] be a subsequence of [1, . . . , n] with complement [jm+1 , . . . , jn ], and let ̺1 , . . . , ̺n−m ∈ (̺ ,...,̺ ) {1, −1}. For f ∈ R[x1 , . . . , xn ], write f[j11,...,jmn−m for the polynomial over R obtained from f by substi] (̺ ,...,̺

tuting xjm+i = ̺i ηjm+i for i = 1, . . . , n − m. For a finite subset H of R[x1 , . . . , xn ], write H[j11,...,jmn−m ] (̺ ,...,̺

)

)

for the set {h[j11,...,jmn−m | h ∈ H}. ] The following proposition is useful for the design of our algorithm. Proposition 3.15. Let f ∈ R[x1 , . . . , xn ], let H be a nonempty finite subset of R[x1 , . . . , xn ] such that Qn ZeroR (H) 6= ∅, and denote by µ the minimum of f on i=1 [−ηi , ηi ]R ∩ ZeroR (H). If µ ∈ / R, then for

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some subsequence [j1 , . . . , jm ] of [1, . . . , n − 1] and certain ̺1 , . . . , ̺n−m ∈ {1, −1},   m Y (̺1 ,...,̺n−m ) (̺1 ,...,̺n−m ) µ = inf VR f[j1 ,...,jm ] : ] − ηji , ηji [R ∩ZeroR (H[j1 ,...,jm ] ) . i=1

(̺ ,...,̺

)

(̺ ,...,̺

)

In particular, µ ∈ EValueR (f[j11,...,jmn−m : H[j11,...,jmn−m ). ] ] Qn Proof. Since µ is the minimum of f on i=1 [−ηi , ηi ]R ∩ ZeroR (H), there exists an (α1 , . . . , αn ) in Qn [−η i , ηi ]R ∩ ZeroR (H) such that f (α1 , . . . , αn ) = µ. Put i=1 Γ := {j | 1 6 j 6 n, and − ηj < αj < ηj }, and denote by m the number of subscripts in Γ. Then 0 6 m 6 n. Suppose that m = 0. Then Γ = ∅, and αi = ̺i ηi for some ̺i ∈ {1, −1}, i = 1, . . . , n. In this case, α1 , . . . , αn are algebraically independent over R. So we have h(α1 , . . . , αn ) 6= 0 for every h ∈ H, a contradiction. Hence m > 1. Qn Suppose that m = n. Then µ is the minimum of f on i=1 ] − ηi , ηi [R ∩ZeroR (H). In this case, we have µ ∈ EValueR (f : H). By Remark 3.9, µ is algebraic over R. Since R is a real closed field, R is algebraically closed in R. So we have µ ∈ R, a contradiction. Hence m < n. Now assume that Γ = {j1 , . . . , jm }, where 1 6 j1 < · · · < jm 6 n. Denote by [jm+1 , . . . , jn ] the complement of [j1 , . . . , jm ] in [1, . . . , n]. Then, −ηji < αji < ηji for i = 1, . . . , m, but αjm+i = ̺i ηjm+i for some ̺i ∈ {1, −1}, i = 1, . . . , n − m. (̺ ,...,̺ ) Qm (̺ ,...,̺ ) It is easy to see that µ = inf VR (f[j11,...,jmn−m : i=1 ]−ηji , ηji [R ∩ ZeroR (H[j11,...,jmn−m )). It follows that ] ]   m Y (̺1 ,...,̺n−m ) (̺1 ,...,̺n−m ) : ] − ηji , ηji [R ∩ ZeroR (H[j1 ,...,jm ] ) . µ ∈ EValueR f[j1 ,...,jm ] i=1

By Proposition 3.2(3), we have (̺ ,...,̺

)

(̺ ,...,̺

)

µ ∈ EValueR (f[j11,...,jmn−m : ZeroR (H[j11,...,jmn−m )), ] ] (̺ ,...,̺

)

(̺ ,...,̺

)

i.e., µ ∈ EValueR (f[j11,...,jmn−m : H[j11,...,jmn−m ). This completes the proof. ] ] Lemma 3.16. Let ρ be a polynomial over R[η1 , . . . , ηn ] in one variable t, let α be a root of ρ in A, and u(t) the leading coefficient of ρ as a polynomial over R[t] in variables η1 , . . . , ηn with respect to the lexicographic order η1 ≺ · · · ≺ ηn . Then π(α) is a root of u(t) in R. Proof.

Represent ρ in the form ρ = u(t)η1d1 · · · ηndn + u1 (t)η1d11 · · · ηnd1n + · · · + ur (t)η1dr1 · · · ηndrn ,

where ui (t) ∈ R[t], i = 1, . . . , r, and u(t) is the leading coefficient of ρ as a polynomial over R[t] in variables η1 , . . . , ηn with respect to the lexicographic order η1 ≺ · · · ≺ ηn . By the above representation, we have ρ(α) = 0, u(α) + µ1 + · · · + µr = d1 η1 · · · ηndn d

where µi := ui (α) ·

d

η1 i1 ···ηnin d dn η1 1 ···ηn

, i = 1, . . . , r.

By the definition of the ordering of R, it is easy to prove that µi ∈ M for i = 1, . . . , r. Hence π(µi ) = 0 for i = 1, . . . , r. By Proposition 3.2(1), we have u(π(α)) = π(u(α)) = π(u(α) + µ1 + · · · + µr ) = π(0) = 0. This completes the proof.

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Now, we proceed to establish the algorithm for the general case. Let F be a computable ordered subfield of R, e.g., F = Q. It is easy to see that F(η1 , . . . , ηn ) is a computable ordered subfield of R with respect to its inherited ordering. In what follows, for a subsequence ̟ := [j1 , . . . , jm ] of [1, . . . , n] with complement [jm+1 , . . . , jn ] and f ∈ R[x1 , . . . , xn ], write f̟ for the polynomial over F[η1 , . . . , ηn ] obtained from f by substituting xjm+i = ηjm+i for i = 1, . . . , n − m. For a finite subset H of R[x1 , . . . , xn ], write H̟ for the set {h̟ | h ∈ H} of polynomials over F[η1 , . . . , ηn ]. Algorithm 3 (Catching the constrained infimums of polynomials). Structure: A computable ordered subfield F of R. Input: A polynomial f ∈ F[x1 , . . . , xn ] and a finite subset H of F[x1 , . . . , xn ]. Output: A finite set U of univariate polynomials in F[t] such that • inf V(f : H) is a root of some polynomial in U whenever inf V(f : H) 6= ±∞. Procedure: Step 1. By Algorithm 2, compute a finite set U0 of univariate polynomials in F[t] such that • inf V(f : H) is a root of some polynomial in U0 whenever inf V(f : H) is attained. Step 2. For every subsequence ̟ of [1, . . . , n], implement the following computations: (1) By Algorithm 2 and Remark 3.9, compute a finite set W̟ of polynomials over F[η1 , . . . , ηn ] in one variable t such that • α is a root of some polynomial in W̟ for every α ∈ EValueR (f̟ : H̟ ). (2) For every w ∈ W̟ , extract the leading coefficient of w as a polynomial over F[t] in variables η1 , . . . , ηn with respect to the lexicographic order η1 ≺ · · · ≺ ηn , and denote by U̟ the set of such extracted leading coefficients. S Step 3. RETURN (U := U0 ∪ ( ̟ U̟ )), where ̟ runs over all the subsequences of [1, . . . , n].

Proof of correctness. Assume inf V(f : H) 6= ±∞. Then ZeroR (H) 6= ∅; otherwise inf V(f : H) = +∞. By Step 1, it is clear that inf V(f : H) is a root of some polynomial in U when inf V(f : H) is attained. Now assume that inf V(f : H) is not attained. According to Proposition 3.11, f has its minimum µ Qn on i=1 [−ηi , ηi ]R ∩ ZeroR (H). By Proposition 3.13, π(µ) = inf V(f : H), but µ ∈ / R. According to Proposition 3.15, (̺ ,...,̺ ) (̺ ,...,̺ ) ) : H[j11,...,jmn−m µ ∈ EValueR (f[j11,...,jmn−m ] ] for some subsequence [j1 , . . . , jm ] of [1, . . . , n] and certain ̺1 , . . . , ̺n−m ∈ {1, −1}. Let [jm+1 , . . . , jn ] be the complement of [j1 , . . . , jm ] in [1, . . . , n]. Obviously, there exists an Fautomorphism φ of the ring F[η1 , . . . , ηn ][t, x1 , . . . , xn ] such that (ηj1 · · · ηjn , t, x1 , . . . , xn ) → (ηj1 · · · ηjm , ̺1 ηjm+1 · · · ̺n−m ηjn , t, x1 , . . . , xn ). (̺ ,...,̺

)

(̺ ,...,̺

)

Put ̟1 := [j1 , . . . , jm ]. Then φ(f̟1 ) = f[j11,...,jmn−m , and φ(H̟1 ) = H[j11,...,jmn−m . By the computations ] ] in Step 2, the finite subset W̟1 , obtained by Algorithm 2 from f̟1 and H̟1 , of F[η1 , . . . , ηn ][t] satisfies the following condition: • α is a root of some polynomial in W̟1 for every α ∈ EValueR (f̟1 : H̟1 ). By the design of Algorithm 2, it is easy to see that φ(W̟1 ) is just the finite subset of F[η1 , . . . , ηn ][t], (̺ ,...,̺ ) (̺ ,...,̺n−m ) which is by Algorithm 2 obtained from f[j11,...,jmn−m and H[j1 , . . . , jm ] 1 . So the following con] dition is satisfied: (̺ ,...,̺ ) (̺ ,...,̺ ) • α is a root of some polynomial in φ(W̟1 ) for every α ∈ EValueR (f[j11,...,jmn−m : H[j11,...,jmn−m ). ] ] Since (̺ ,...,̺ ) (̺ ,...,̺ ) µ ∈ EValueR (f[j11,...,jmn−m : H[j11,...,jmn−m ), ] ] µ is a root of φ(w) for some w ∈ W̟1 . Let u(t) be the leading coefficient of w as a polynomial over F[t] in variables η1 , . . . , ηn with respect to the lexicographic order η1 ≺ · · · ≺ ηn . Then u(t) ∈ U̟1 ⊆ U . It is easy to see that either u(t) or −u(t) is the leading coefficient of φ(w) as a polynomial over F[t] in variables η1 , . . . , ηn with respect to the lexicographic order η1 ≺ · · · ≺ ηn . By Lemma 3.16, we have either u(π(µ)) = 0 or −u(π(µ)) = 0, and u(π(µ)) = 0. So inf V(f : H) is a root of the polynomial u(t) in U . The correctness of Algorithm 3 is verified.

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Two examples

In this section, we use the Algorithms 2 and 3 to treat two examples with the aid of the computer algebraic system Maple and the software wsolve adapted to Wu’s algorithm. For the details of Maple and wsolve, refer to references [8, 25], respectively. As an illustration of Algorithm 2, we proceed to investigate the following example. Example 4.1. Let f = x6 + y 6 + z 6 + 3x2 y 2 z 2 − x2 (y 4 + z 4 ) − y 2 (x4 + z 4 ) − z 2 (y 4 + x4 ), and H = {x4 + y 4 + z 4 − 1}. Find a finite set of univariate polynomials such that inf V(f : H) is a root of some univariate polynomial. Observe that the feasible region ZeroR (H) is a closed and bounded subset of R3 . So inf V(f : H) is attained. Process of computing. According to Algorithm 2, we proceed to perform the following computations: (1) With respect to the lexicographic order x ≻ y ≻ z, by the Wu’s algorithm for Zero-Decomposition Theorem II, compute a triangular decomposition of H with seperants as follows: C1 := [x4 + y 4 + z 4 − 1], C3 := [x, y, z − 1],

C2 := [x, y 4 + z 4 − 1],

C4 := [x, y, z + 1],

C5 := [x, y, z 2 + 1].

(2) For i = 1, . . . , 5, create a finite subset Ui of Q[t] from Ci in the following manner. (2.1) According to Definition 3.4, the set of Lagrangian polynomials of f and C1 is computed as follows: L(f, C1 ) = {40x3 y 5 + 24x5 yz 2 − 40x5 y 3 − 8x7 y − 8x3 yz 4 − 24y 5 xz 2 + 8y 7 x + 8y 3 xz 4 , 40x3 z 5 + 24x5 y 2 z − 40x5 z 3 − 8x3 zy 4 − 8x7 z − 24z 5 xy 2 + 8z 3 xy 4 + 8z 7 x}. With respect to the lexicographic order x ≻ y ≻ z, by the Wu’s algorithm for Zero-Decomposition Theorem I, compute a triangular decomposition of C1 ∪ L(f, C1 ) as follows: D1 := [x, y 4 + z 4 − 1],

D2 := [x, −1 + 3z 4 + 3yz 3 , 9z 8 − 9z 4 + 1],

D3 := [x, 1 − 3z 4 + 3yz 3 , 9z 8 − 9z 4 + 1], D5 := [1 + 6y 2 x2 , 36y 8 − 36y 4 + 1, z], 4

D8 := [x − y, 2y − 1, z], D10 := [x − z, y, −1 + 2z 4 ],

D6 := [x − 1, y, z], 4

D7 := [x + 1, y, z],

D9 := [x − y, 3z + 6y z − 1, 27z 8 − 24z 4 + 1],

2 2

D11 := [1 + 3x2 z 2 − 3z 4 , z + y, −15z 4 + 27z 8 + 1],

D12 := [x − z, z + y, −1 + 3z 4 ], D14 := [x + z, y, −1 + 2z 4 ],

D4 := [1 + 6x2 z 2 , y, 36z 8 − 36z 4 + 1],

D13 := [x − z, −3z 4 + 3y 2 z 2 + 1, −15z 4 + 27z 8 + 1],

D15 := [x + z, y − z, −1 + 3z 4 ],

D17 := [x + z, −3z 4 + 3y 2 z 2 + 1, −15z 4 + 27z 8 + 1], D19 := [y + x, 3z 4 + 6y 2 z 2 − 1, 27z 8 − 24z 4 + 1], D21 := [1 + 3x2 z 2 − 3z 4 , y − z, −15z 4 + 27z 8 + 1],

D16 := [x + z, z + y, −1 + 3z 4 ],

D18 := [y + x, 2y 4 − 1, z],

D20 := [x2 + 1, y, z], D22 := [x − z, y − z, −1 + 3z 4 ].

According to Definition 2.11, implement the algorithm CUP as follows: By introducing a new variable t, we get RvRes(t − f, D1 ) = (t2 − 4tz 6 + 2tz 2 + 8z 12 − 12z 8 + 6z 4 − 1)2 . Regarding RvRes(t−f, C) as a polynomial over Q[t] in one variable z, all the coefficients of RvRes(t−f, C) are as follows: {−192, 64, 240, −96t, −64t, 128t, −160 + 32t2 , −12 + 16t2 , 60 − 40t2 , −4t + 4t3 , 32t − 8t3 , 1 − 2t2 + t4 }. Obviously, the greatest common divisor of the coefficients of RvRes(t − f, C) is 1. So we put u1 (t) := 1. Furthermore, we get RvRes(t − f, D2 ) = 150094635296999121(27t2 − 5)4 . So we may put u2 (t) := 27t2 − 5. Likewise, by computing RvRes(t − f, Di ) for i = 3, . . . , 22, we obtain such polynomials as follows:

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27t2 − 5, 27t2 − 32, t − 1, t − 1, t, 243t4 − 423t2 + 1, t, t, 243t4 − 423t2 + 1, 243t4 − 423t2 + 1, t, t, t, 243t4 − 423t2 + 1, t, 243t4 − 423t2 + 1, t + 1, 27t2 − 32, 243t4 − 423t2 + 1, t. The obtained non-constant polynomials in one variable t constitute such a finite set as follows: U1 := {t, t − 1, t + 1, 27t2 − 32, 27t2 − 5, 243t4 − 423t2 + 1}. (2.2) For the chain C2 , by the computations as in (2.1), we may obtain such a finite set of polynomials over Q in one variable t as follows: U2 := {t, t − 1, t + 1, 27t2 − 32}. (2.3)–(2.5) Observe that Ci is complete for i = 3, 4, 5. By the algorithm CUP adapted to (f, Ci ) (in Definition 2.11) for i = 3, 4, 5, we obtain such finite sets of polynomials over Q in one variable t as follows: U3 := {t − 1}, U4 := {t − 1}, U5 := {t + 1}. Put U :=

S

16k65

Uk . Then U = {t, t − 1, t + 1, 27t2 − 32, 27t2 − 5, 243t4 − 423t2 + 1} is required.

Example 4.2. Let f = 2y 2 z 2 +x2 y 2 +2x2 +y 4 +2xy +3y 2 , and H = {x2 +2y 2 z −z 2 −xy +1, x+y −z}. Find a finite set of univariate polynomials such that inf V(f : H) is a root of some univariate polynomial whenever inf V(f : H) 6= ±∞. Process of computing. Observe that all the nonempty subsequences of [x, y, z] are [x, y, z], [x, y], [x, z], [y, z], [x], [y] and [z]. According to Algorithm 3, we implement the following computations: (1) With respect to the lexicographic order x ≻ y ≻ z, by the Wu’s algorithm for Zero-Decomposition Theorem II, compute a triangular decomposition of H with seperants as follows: C1 := [x + y − z, 2y 2 z + 1 + 2y 2 − 3yz],

C2 := [3x + 2, 1 + 3y, z + 1],

C3 := [36xz + 36x − 41z − 32, 4yz + 4y − 3z, 9z 2 − 8z − 8]. (1.1) For i = 1, 2, 3, create a finite subset Ui of Q[t] from Ci in the following manner. (1.1.1) According to Definition 3.4, the set of Lagrangian polynomials of f and C1 is computed as follows: L(f, C1 ) = {8y 3 z 2 + 16y 3 z + 8xy 3 z + 2xy 3 − 6xy 2 z + 16zxy + 10xy − 12xz + 8y 2 z + 20y 2 − 6yz + 4xy 4 + 4xy 2 − 8y 3 − 4x2 y 3 + 6x2 y 2 − 8y 5 + 12y 4 }. With respect to the lexicographic order x ≻ y ≻ x, by the Wu’s algorithm for Zero-Decomposition Theorem I, compute a triangular decomposition of C1 ∪ L(f, C1 ) as follows: D1 := [325z 5x − 301z 6 + 86z 6 x − 86z 7 + 335z 4x − 221z 5 + 404z 3x − 201z 4 + 164xz 2 − 45z 3 + 22xz + 50z 2 + 4x + 10z + 2, 325yz 5 + 86yz 6 − 24z 6 + 335yz 4 − 114z 5 + 404yz 3 − 203z 4 + 164yz 2 − 119z 3 + 22yz − 72z 2 + 4y − 14z − 2, 1260z 9 + 3290z 8 − 1890z 7 − 2328z 6 − 5326z 5 − 5537z 4 − 1391z 3 − 432z 2 − 44z − 6]. According to Definition 2.11, implement the algorithm CUP as follows: By introducing a new variable t, we get RvRes(t − f, D1 ) = cu1 (t), where c is a positive integer, and u1 (t) := 5308416t9 − 1777256960t8 − 7231628064t7 + 27569302304t6 + 178147099540t5 + 234789724229t4 − 142470441628t3 − 570694202594t2 − 463848310336t − 125289938395. So we may put U1 := {u1 (t)}. (1.1.2) Observe that Ci is complete for i = 2, 3. Implementing the algorithm CUP adapted to (f, Ci ) for i = 2, 3, we obtain such finite sets of polynomials over Q in one variable t as follows: U2 := {81t − 158}, U3 := {6561t2 − 70308t + 186574}.

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S (1.1.3) Put U[x,y,z] := 16k63 Uk . Then U[x,y,z] = {u1 (t), 81t − 158, 6561t2 − 70308t + 186574}. (2) With respect to the lexicographic order y ≺ x, by the Wu’s algorithm for Zero-Decomposition Theorem II, compute a triangular decomposition of H with seperants as follows: C4 := [x + y − z, 2y 2z + 1 + 2y 2 − 3yz]. Regarding z as a constant parameter, C4 is a complete chain. By introducing a new variable t, we get RvRes(t − f, C4 ) = w(t, z), where w(t, z) := 16 + 48z + 32t + 34z 2 − 438z 4t − 268z 5 t + 64t2 z + 16t2 + 96t2 z 2 + 64t2 z 3 − 172z 6t + 16t2 z 4 + 112tz + 124tz 2 − 136tz 3 − 172z 3 − 302z 4 + 12z 5 + 456z 6 + 280z 7 + 280z 8. Obviously, the greatest common divisor of all the coefficients of w(t, z), regarded as a polynomial over Q[t] in one variable z, is 1. So we may put U[x,y] := ∅. (3),(4) Implementing the same computations as in (2), we may get U[x,z] = U[y,z] = ∅. (5)–(7) Regard the polynomials in H as polynomials over Q[y, z] in one variable x. Then the triangular decomposition of H with seperants is the empty set ∅. So we put U[x] := ∅. S Likewise, we may get U[y] = U[z] = ∅. Put U := ̟ U̟ , where ̟ runs over all the nonempty subsequences of [x, y, z]. Then U = {u1 (t), 81t − 158, 6561t2 − 70308t + 186574} is required. With the aid of the computer algebraic system Maple 15 and the software wsolve, Algorithms 2 and 3 have been made into a general program, named ConstrainedInfimum, to treat the equality-constrained minimization of polynomials with rational coefficients. The software ConstrainedInfimum can be found in [29]. The software ConstrainedInfimum contains the principal functions as follows: • ConstrainedMin for implementing Algorithm 2 when the input polynomials have their rational coefficients. • ConstrainedInf for implementing Algorithm 3 when the input polynomials have their rational coefficients. By using the software ConstrainedInfimum, Examples 4.1 and 4.2 were treated on a Pentium IV computer with 128MB RAM. Calling ConstrainedMin(f, H, [x, y, z]) for f and H in Example 4.1 and ConstrainedInf(f, H, [x, y, z]) for f and H in Example 4.2, we may obtain the same results as above. The respective CPU times are 0.734s and 0.593s. In order to compare our algorithm with the method [27] (or [28, Subsection 5.5]), we treat Example 4.1 according to the proof of Finite Kernel theorem [27] (or the relevant description in [28, Subsection 5.5]). Since the function evalue in the software wsolve is devised for the Zero-decomposition theorem II, we call evalue({t − f, h}, [x, y, z, t], {}), where f = x6 + y 6 + z 6 + 3x2 y 2 z 2 − x2 (y 4 + z 4 ) − y 2 (x4 + z 4 ) − z 2 (y 4 + x4 ), h = x4 + y 4 + z 4 − 1 and t is a new variable. However, there is no result in 500 seconds.

5

Conclusions

In this paper, by decomposing a finite set of polynomials into triangular chains and computing the socalled revised resultants, we give an algorithm to create a finite set of univariate polynomials such that the finite constrained infimum of a polynomial is a root of some univariate polynomial. Our algorithm does not require the attainability of the constrained infimum or any assumption on the constraint polynomials. However, the following problems remain unsolved for an f ∈ R[x1 , . . . , xn ] and a finite subset H of R[x1 , . . . , xn ]: (1) How to compute the accurate value of the constrained infimum inf V(f : H)? (2) How to decide whether or not inf V(f : H) is attained when inf V(f : H) 6= ±∞? Based on the results and algorithms established in this paper, these problems will be investigated in a forthcoming article.

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Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 11161034). The authors are very grateful to the referees for their valuable suggestions that helped to improve this paper.

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