ideal projectors are the limits of Lagrange projectors? The results of this paper answer the question in the sense that for every ideal projector we prescribe.
On the Limits of Lagrange Projectors Boris Shekhtman Abstract. This article addresses a question of Carl de Boor (cf. [3]): What ideal projectors are the limits of Lagrange projectors? The results of this paper answer the question in the sense that for every ideal projector P we prescribe …nitely many computations that determine whether the projector P is a limit of Lagrange projectors.
1. Introduction The main goal of this article is to answer the following question of Carl de Boor (cf. [3]): What ideal projectors are the limits of Lagrange projectors? Let C[x] = C[x1 ; : : : ; xd ] be the ring of polynomials in d variables with complex coe¢ cients. Definition 1.1. (cf. [1]) A linear idempotent map P on C[x] is called an ideal projector if ker P is an ideal in C[x]. A simple but extremely useful description of ideal projectors is given by de Boor’s formula (1.0): Theorem 1.2. (cf. [2]) A linear mapping P : C[x] ! C[x] is an ideal projector if and only if the equality (1.0)
P (f g) = P (f P g)
holds for all f; g 2 C[x]. A standard example of a …nite-dimensional ideal projector is a Lagrange projector, i.e., a linear projector P for which P f is the unique element in its range that agrees with f at a certain …nite set Z in Cd . To put it in another way, P is a Lagrange projector if and only if there exists a …nite set of points Z Cd such that #Z = dim ran P and (P f )(z) = f (z) for all z 2Z and for all f 2 C[x]. It is an ideal projector, for its kernel consists of exactly those polynomials that vanish on Z, i.e., it is the radical ideal whose variety is Z. Definition 1.3. A (…nite-dimensional) projector P is a Hermite projector if there exists a sequence of Lagrange projectors Pn onto ran P such that (Pn f )(z) ! (P f )(z) for every f 2 C[x] and every z 2 Cd . 1991 Mathematics Subject Classi…cation. 41A05, 41A10, 41A35, 41A63, 14C05. Key words and phrases. Ideal Projector, Lagrange Projector, A¢ ne Variety, Rational Parametrization. 1
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BORIS SHEKHTM AN
Clearly (cf. [2], [3]), a Hermite projector, being the limit of Lagrange projectors, is an ideal projector. In one variable, every ideal projector is a Hermite projector. It was shown in [9] that the same is true for d = 2, while for every d 3, there exist ideal projectors that are not Hermite. Hence one is interested in a characterization of those ideal projectors that are. A question of characterization is often open-ended. The satisfaction with the answer depends on the terms in which the answer is expected and presented. The results of this paper answers the question in the sense that for every ideal projector P we prescribe …nitely many steps of symbolic computations that determine whether the projector P is Hermite. The bad news is that “…nitely many” is still far too many steps for my computer to handle even in a simplest non-trivial case. This is not too surprising, since such an example must occur for the projectors in three variables onto polynomials of degree 2, that is for the projectors onto a 10-dimensional space that, non-trivially, depend on 100 parameters. The description of Hermite projectors presented in this paper follows the following steps: 1) Given an N -dimensional subspace G C[x] we use (1.0) to parametrize the family PG of all ideal projectors onto G by an a¢ ne algebraic variety WG CmN where m depends on G. That is, every mN -vector w 2 WG gives rise to a projector Pw 2 PG and every projector P 2 PG corresponds to a vector wP 2 WG with PwP = P and wPw = w. The coordinates of wP can be obtained by applying the projector P to a given …nite sequence of functions (h1 ; : : : ; hm ) that depends explicitly on G. 2) We show that the family (1.1)
HG := fw 2 WG : Pw is Hermiteg
WG
of those w 2 WG that give rise Hermite projectors admits an explicit rational parametrization (in terms of polynomials in G) and is an (irreducible) subvariety of WG . 3) We describe a rational parametrization of the variety HG that yields a Groebner basis ff1 ; : : : ; fs g for the ideal (1.2)
I(HG ) = ff 2 C[w] := C[w1 ; : : : ; wmN ] : f (w) = 0; 8w 2 HG g
in …nitely many steps. 4) We now start with an ideal projector P 2 PG and compute wP 2 WG . Next, compute the values: f1 (wP ); : : : ; fs (wP ). If (1.3)
f1 (wP ) = f2 (wP ) =
= fs (wP ) = 0
then P is Hermite; if not then it is not. We note that it may, and does, happen that HG = WG for some G, in which case (1.3) holds for every ideal projector P 2 PG and hence every ideal projector onto G is Hermite. The main argument of the paper is the proof that HG is the Zariski closure of the family (1.4)
LG := fw 2 WG : Pw is Lagrangeg.
This argument is where some machinery from algebraic geometry is used.
LAGRANGE LIM ITS
3
2. Preliminaries We will need to recall a few basic facts from algebraic geometry, most of which can be found in [4] and [5]. Given a subset S C[x] we use hSi to denote the ideal generated by S. For an ideal I C[x] we use Z(I) := fx 2 Cd : f (x) = 0; 8f 2 Ig
(2.1)
to denote the associated a¢ ne variety in Cd . A subset V Cd is an a¢ ne variety if and only if there exists an ideal I C[x] such that V = Z(I). One such ideal is the corresponding radical ideal (2.2)
I(V) := ff 2 C[x] : f (x) = 0; 8x 2 Vg.
In fact, for any subset U (2.3)
Cd we can de…ne the ideal
I(U) := ff 2 C[x] : f (x) = 0; 8x 2 U g.
Hence U is an a¢ ne variety if and only if Z(I(U)) = U. Every a¢ ne variety V Cd comes with an intrinsic "Zariski topology", de…ned by taking the closed subsets to be its a¢ ne subvarieties. Thus for every subset U V, one de…nes the Zariski closure of U to be the least subvariety (Zariski closed set) that contains U. A variety V Cd is called irreducible if V is not a union of two proper subvarieties of V. If U is a Zariski open subset of an irreducible variety V, then the Zariski closure of U is V. Indeed, since U is Zariski open, its complement VnU is closed and therefore a subvariety of V. If the Zariski closure Ue of U is di¤erent from V then V = (VnU)[ Ue is a union of two proper subvarieties of V which is a contradiction. Of course, as a subset of Cd , an a¢ ne variety V Cd inherits the Euclidean d topology from C , which is stronger (…ner) than the Zariski topology. As a common zero locus of polynomials, every a¢ ne variety is closed in the Euclidean topology as well as the Zariski topology. The converse is not true: The Zariski closure of a set U Cd can be much larger than its Euclidean closure. However, for irreducible varieties there are instances when the two coincide. Theorem 2.1. (cf. [7], Theorem 1, p. 82) If U is a non-empty Zariski open subset of an irreducible variety V, then the Euclidean closure U of U coincides with its Zariski closure, hence U = V. We will now address the issues of the rational parametrization of a variety: Given an n-tuple of rational functions (2.4)
(
pj (x) : pj (x); qj (x) 2 C[x]; j = 1; :::; n) qj (x)
we follow [4], p.129 and de…ne a rational map Cd nZ(hq1 ; : : : ; qn i) into Cn given by (2.5)
(x) = (
: Cd ! Cn as the map from
p1 (x) pn (x) ;:::; ) 2 Cn . q1 (x) qn (x)
We say that a variety V Cn admits a rational parametrization (by Cd ) if there exists a rational map : Cd ! Cn such that V is the smallest variety in Cn
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BORIS SHEKHTM AN
that contains the image of . That is, V is the Zariski closure of the image of . In this case we will also say that V is parametrized by . Cn is a variety that
Theorem 2.2. (cf. [4], Proposition 6, p. 197): If V admits a rational parametrization, then V is irreducible.
Since V is a variety, the Hilbert basis theorem tells us that the ideal I(V) has a …nite number of generators. It is remarkable that the use of Groebner bases makes it possible to …nd these generators in …nitely many steps of symbolic computations as described in [4], pages 129–131: Given the rational parametrization of V, this implicitization is accomplished by creating a Groebner basis for the ideal * + n Y J := ti qi pi ; 1 tn+1 qj : i = 1; : : : ; n j=1
in the space C[t1 ; : : : ; tn ; tn+1 ; x1 ; : : : ; xd ] with respect to the lexicographic order with xd xd 1 x1 tn+1 tn t1 . The d-th elimination ideal J(d) is generated by those polynomials f1 ; : : : ; fs in the Groebner basis that contain t1 ; : : : ; tn+1 only. The Closure Theorem (cf. [4], Theorem 2, p. 130) ensures that I(V) = J(d) = hf1 ; : : : ; fs i .
Theorem 2.3. Let V be a variety in Cn with a given rational parametrization (2.4). Then it requires only …nitely many steps to compute a sequence of polynomials f1 ; : : : ; fs 2 C[x1 ; : : : ; xd ] such that I(V) = hf1 ; : : : ; fs i. Finally we recall one simple fact about the discriminants of polynomials in one variable: Proposition 2.4. (cf. [5], Exercise 8 a., p. 340) For every N a polynomial 'N 2 C[a0 ; : : : ; aN 1 ] such that the polynomial p(x) = a0 + a1 x + ::: + aN
N 1 1x
has N distinct zeroes in C if and only if 'N (a0 ; : : : ; aN
0 there exists
+ xN 1)
6= 0.
3. Limits of Lagrange Projectors From this point on, we …x an N -dimensional subspace G C[x], a basis fg1 ; : : : ; gN g of G and a sequence (h1 ; : : : ; hm ) comprising all polynomials in the set f1g [ fxi gk : i = 1; : : : ; d; k = 1; : : : ; N g that are not in G. Let PG be the family of all ideal projectors onto G. We start by describing a variety WG that parametrizes the family PG : It follows from de Boor’s formula (1.0) that every ideal projector P 2 PG is completely determined by its m values (polynomials) (3.1)
P hj =
N X
k=1
and the set (3.2)
fhj
wj;k gk 2 G; j = 1; : : : ; m,
N X
k=1
wj;k gk : j = 1; : : : ; mg
LAGRANGE LIM ITS
5
is a basis for the ideal ker P . Equivalently, with every P 2 PG we can associate
wP := (wj;k : j = 1; : : : ; m; k = 1; : : : ; N ) 2 CmN
(3.3)
that completely determines the projector P . Moreover, (3.4)
Pf =
N X
qm;f (wP )gm
m=1
where qm;f are …xed polynomials in the mN variables wP . Indeed, (3.4) holds for f = 1 and f = xi gk ; i = 1; : : : ; d; k = 1; : : : ; N . Inductively, assume that (3.4) holds for all monomials g of degree n and let f be a monomial of degree n + 1. Then f = xi g for some i and, by (1.0) we have P (f )
= P (xi g) = P (xi P (g)) = P (xi (
N X
qm;g (wP )gm ))
m=1
=
N X
qm;g (wP )P (xi gm ) =
m=1
m=1
=
N X
k=1
"
N X
qk;xi gm (wP )
N X
m=1
qm;g (wP ) #
N X
qk;xi gm (wP )gk
k=1
qm;g (wP ) gk .
PN Since qk;xi gm (wP ) m=1 qm;g (wP ) is a polynomial, hence (3.4) holds for monomials and thus for all f 2 C[x]. De…ne the set (3.5)
WG := fw 2 CmN : w = wP for some P 2 PG g
in CmN of those sequences w that determine ideal projectors onto G. We claim that WG is an a¢ ne variety. Proposition 3.1. The set WG CmN is an a¢ ne variety, the formulas (3.1) determine a one-to-one correspondence between PG and WG , i.e., for every w 2 WG there exists a unique projector Pw 2 PG with wPw = w as well as PwP = P for every P 2 PG . Moreover, wn ! w, (wn 2 WG ) if and only if Pwn f ! Pw f for every f 2 C[x]. In this case Pw 2 PG . Proof. We only need to prove that WG is an a¢ ne variety. By de Boor’s formula (1.0), a given w 2 CmN de…nes an ideal projector by (3.1) if and only if the identity (3.6)
P (f1 P f2 ) = P (f3 P f4 )
holds for all f1 ; f2 ; f3 ; f4 2 C[x] such that f1 f2 = f3 f4 . By (3.4), each identity (3.6) translates into a system of polynomial equations in w 2 CmN . Thus WG , as the zero locus of these equations, is an a¢ ne variety. Pw and
Next, we wish to determine those w 2 CmN that correspond to ideal projectors that are Lagrange. Let LG denotes the set of all Lagrange projectors onto G
(3.7)
LG := fw 2 WG : Pw 2 LG g
WG
be the subset of WG that corresponds to Lagrange projectors.
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BORIS SHEKHTM AN
Theorem 3.2. LG is a Zariski open set. by
Proof. With every w 2 WG we associate linear operators Mj;w de…ned on G
Mj;w g := Pw (xj g); j = 1; :::; d. ~ and let Mj;w be the N N matrix of the operator Mj;w in the basis fg1 ; : : : ; gN g ~ j;w ; j = 1; :::; d) is a sequence of pairwise of G. It is well-known (cf. [2]) that (M ~ j;w ; j = commuting matrices and Pw is a Lagrange projector if and only if (M 1; :::; d) are simultaneously diagonalizable, i.e., there exists a basis f~ g1 ; : : : ; g~N g in G consisting of eigenvectors of Mj;w : Mj;w g~k = zj;k g~k , j = 1; :::; d; k = 1; :::; N for some zj;k 2 C. In this case the projector Pw interpolates at sites zk := (zj;k ; j = 1; :::; d ) 2 Cd . ~ 1;w are polynomials in w. Let By (3.4) the entries of the matrix M ~ 1;w has N distinct eigenvaluesg. L1G := fw 2 WG : M
~ 1;w are Since the coe¢ cients a0 ; : : : ; aN 1 of the characteristic polynomial of M polynomials in w, it follows that 'N (a0 ; : : : ; aN 1 ) is a polynomial in w and, by Proposition 2.4, L1G is a Zariski open subset of WG . g1 ; : : : ; g~N g be a basis in G consisting We now need to show that L1G LG . Let f~ of eigenvectors M1;w . Then each g~k corresponds to the eigenvalue z1;k and spans the one-dimensional eigenspace Gk = ker(z1;k IG M1;w ) of M1;w . By commutativity 0 = Mj;w (z1;k IG
M1;w )~ gk = (z1;k IG
M1;w )Mj;w g~k ,
hence Mj;w g~k 2 Gk and thus g~k is an eigenvector of Mj;w for each j = 1; :::; d. ~ j;w ; j = 1; :::; d) are simultaneously diagonalizable and thus L1 In other words (M G parametrizes the set of all Lagrange projectors that interpolate at sites with distinct …rst coordinates. To prove the theorem, it remains to observe that for every Lagrange projector P there exists a change of variables: X = (X1 ; :::; Xd ) Xj =
d X
bmj xm ; j = 1; :::; d,
m=1
such that P interpolates at the sites z1 ; :::; zN with distinct …rst coordinates with respect to variables (X1 ; X2 ; :::; Xd ).Thus LG is a union of open sets which are copies of L1G with respect to di¤erent variables and the union of open sets is open. Let LG be the Euclidean closure of LG . By Proposition 3.1, LG coincides with the set (3.8)
HG := fw 2 WG : Pw is Hermiteg.
Thus to proceed with our program, we need to characterize LG . Since a Hermite projector is an ideal projector, it follows that LG is a subset of WG , however LG is not, a priori, a subvariety of WG . Theorem 3.3. HG is an irreducible subvariety of WG with an explicit rational parametrization (in terms of polynomials (g1 ; : : : ; gN ) and (h1 ; : : : ; hm )). Hence, by Theorem 2.3, there exist polynomials f1 ; : : : ; fs (that depend on the image space
LAGRANGE LIM ITS
7
G only), computable in …nitely many steps, such that an ideal projector P 2 PG is Hermite if and only if f1 (wP ) = = fs (wP ) = 0. Proof. Let VG be the Zariski closure of LG . Since VG is a variety, we …rst establish a rational parametrization for VG , concluding (by Theorem 2.2) that VG is irreducible. We then prove that LG is Zariski open in VG , which implies, by Theorem 2.1, that HG = LG = L~G , thus proving the theorem. We start with an explicit description of a rational map onto LG . For every Z = (z1 ; : : : ; zN ) 2 (Cd )N , we introduce the sequence z = (z1;1 ; : : : ; z1;d ; : : : ; zN;1 ; : : : ; zN;d ) 2 CdN
(3.9)
and the collocation matrix (generalized Vandermonde): (3.10)
VG (z) := (gk (zi ) : k; i = 1; : : : ; N ).
Once again, we have a one-to-one correspondence between LG and ZG := fz 2 CdN : det VG (z) 6= 0g.
(3.11)
We will extend this correspondence to the set WG . For every Z = (z1 ; : : : ; zN ) 2 (Cd )N in ZG de…ne the Lagrange projector PZ 2 LG by requiring that (3.12)
(PZ f )(zi ) = f (zi ); i = 1; : : : ; N , for all f 2 C[x]
or, equivalently, (3.13)
PZ hj (zi ) =
N X
wj;k gk (zi ) = hj (zi ); j = 1; : : : ; m; i = 1; : : : ; N .
k=1
Considering (3.13) as a system of mN equations in the mN unknowns (3.14)
(wj;k ; j = 1; : : : ; m; k = 1; : : : ; N ) := w
and using Cramer’s rule, we have (j;k)
(3.15)
wj;k =
det VG (z) det VG (z)
(j;k)
where VG (z) is obtained from VG (z) by replacing its mN -dimensional (j; k)-th column with the vector (3.16) Since
(hj (zi ) : j = 1; : : : ; m; i = 1; : : : ; N ). (j;k) det VG (z)
is a polynomial in dN variables, the image of the rational map : CdN nZ(hdet VG (Z)i) ! CmN
(3.17) de…ned by
(j;k)
(3.18)
(z) := (
det VG (z) : j = 1; : : : ; m; k = 1; : : : ; N ) det VG (z)
is precisely LG . Hence VG is parametrized by the rational function (3.18) given explicitly in terms of polynomials (g1 ; : : : ; gN ) and (h1 ; : : : ; hm ). By Theorem 2.2 we conclude that VG is irreducible. By Theorem 3.2, LG is a non-empty open set hence, by Theorem 2.1, we conclude that LG = VG .
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BORIS SHEKHTM AN
Remark 3.4. It would be nice to have explicit polynomials f1 ; : : : ; fs with LG = Z(hf1 ; : : : ; fs i), for some simple subspaces G such as the space C
