Sep 16, 2014 - using Fried, Haran and Jarden'article [3]. They show that if k := dimX, D := degX, δ := deg â, then the number Nm := |(X \ V (â))(Fqm )| of the Fqm ...
Tangent Codes Azniv Kasparian, Evgeniya Velikova
∗
arXiv:1409.4583v1 [cs.IT] 16 Sep 2014
Abstract We interpret the finite Zariski tangent spaces Ta (X, Fqm ) to an affine variety n X ⊂ Fq as linear codes. If the degree of X is relatively prime to the characteristic p = charFq , we show that the minimum distance of Ta (X, Fqm ) stabilizes while varying a ∈ X(Fqm ) := X ∩ Fnqm and m ∈ N. The article provides a pseudocode for obtaining the generic minimum distance of a tangent code and points, at which it is attained. On the other hand, any family of linear codes of fixed length, fixed dimension and arbitrary minimum distance is interpolated by the n Fq -Zariski tangent bundle of an affine variety X ⊂ Fq with explicit equations, whose degree is divisible by p = charFq .
A basic problem in the theory of error correcting codes is the construction of linear codes with a priori given length, dimension and minimum distance. That is helpful for the presence of a unique decoding under appropriate upper bound on the number of the perturbed symbols. Except the construction of individual linear codes, one can investigate the stabilization or destabilization of the minimum distance within a family of linear codes of fixed length and dimension. Explicit families enable "to deform" a given linear code into one with the desired minimum distance or to produce a bunch of linear codes with one and a same parameters. In a similar vein, one looks for families of Hamming isometries of linear codes. In order to produce linear codes with given parameters or to construct their Hamming isometries, one makes use of structures, arising from other branches of mathematics. In the early 80’s, Goppa introduced the classical algebraic geometric codes. They consist of values of global holomorphic sections of line bundles over curves. Various geometric properties of the corresponding line bundles estimate the dimension and the minimum distance. Under the presence of extra assumptions, one is able to predict the exact values of these parameters. The present work introduces another application of algebraic geometry to the coding theory. It interprets the finite Zariski tangent spaces to an affine variety X as linear codes. If the degree of X is relatively prime to the characteristic p of the definition field of X, the generic minimum distance d of a finite Zariski tangent space to X is characterized by a global geometric property of X. We provide an effective procedure in a form of a pseudo-code, obtaining d and points a ∈ X, at which d is attained. On the other hand, an explicit construction of an affine variety X, whose ∗
This research is partially supported by Contract 101 / 19.04.2013 and Contract 015 / 9.04.2014 with the Scientific Foundation of the University of Sofia
degree is divisible by p, illustrates the possibility for incorporating codes of arbitrary minimum distance within a single Zariski tangent bundle. Here is a synopsis of the paper. The first section characterizes the generic minimum n distance of a tangent code Ta (X, Fqm ) to an affine variety X ⊂ Fq , defined over Fq , whose degree is relatively prime to p = charFq . More precisely, subsection 1.1 recalls the notion of a Zariski tangent space to an affine variety. The next subsection 1.2 translates the minimum distance d of a linear code in terms of its punctures at d− 1 or d coordinates. This description becomes handy for the characterization of the generic minimum distance of a tangent code Ta (X, Fqm ) to X, as far as the puncturings of Ta (X, Fqm ) coincide with the differentials of the associated puncturings of X at a ∈ X. The same subsection 1.2 introduces the genericity index d − 1 of an irreducible affine n variety X ⊂ Fq as the maximal non-negative integer, for which any n−d+1 coordinate functions xj1 + I(X, Fq ), . . . , xjn−d+1 + I(X, Fq ) ∈ Fq (X) contain a transcendence basis of the function field Fq (X) of X over the algebraic closure Fq of the definition field Fq n of X. Let X/Fq ⊂ Fq be an irreducible, (d − 1)-generic affine variety, defined over Fq , whose degree is not divisible by p = charFq . Subsection 1.3 establishes that the generic finite tangent spaces to a complement of a hypersurface in X are of minimum distance d. This involves an analogue of the Implicit Function Theorem, deriving a lower bound on the genericity index of X from a lower bound on the generic minimum distance of a tangent code. A sort of an inverse statement of the Implicit Function Theorem provides a lower bound on the generic minimum distance of a finite Zariski tangent space to X, which follows from a lower bound on the genericity index of X. The second section is devoted to an algorithm for obtaining the generic minimum distance of a tangent code and points, at which it is attained. Its first subsection recalls the Hilbert polynomial of an affine variety and an algorithm for its computing. It sketches one of the well known explicit procedures for decomposition of an affine variety into a union of irreducible components. Subsection 2.2 proposes an easy way for computing the genericity index n d − 1 of an irreducible affine variety X ⊂ Fq , out of the set of all the coordinate transcendence bases of the function field Fq (X) of X over Fq . The third subsection provides an algorithm for obtaining the discriminant locus of all the puncturings of X at d−1 coordinates and explicit points from its complement. Subsection 2.4 synthesizes the algorithms from the previous three subsections in a pseudo-code. The third section illustrates the stabilization, respectively, the destabilization of the minimum distance n within a Zariski tangent bundle to an affine variety X ⊂ Fq , whose degree is not divisible, respectively, is divisible by the characteristic p = charFq . This is done in the corresponding subsections by explicit constructions of the desired affine varieties X by their defining equations. The last, fourth section relates the linear Hamming n isometries of the tangent codes Ta (X, Fqm ) to an affine variety X ⊂ Fq with the differentials of appropriate morphisms of X. Subsection 4.1 provides a pattern for n n construction of a morphism ψ : Fq → Fq , whose differentials restrict to linear Hamming isometries (dψ)a : Ta (X, Fqm ) → Tψ(a) (ψ(X), Fqm ) on the generic Zariski n tangent spaces to the affine varieties X ⊂ Fq , which are not contained in an explicitly n given hypersurface V (ψo ) ⊂ Fq , depending on ψ. Subsection 4.2 realizes arbitrary families of linear Hamming isometries Fnq → Fnq by differentials of appropriate explicit n n morphisms Fq → Fq . It observes also that the Frobenius automorphism Φq of 2
n
an affine variety X ⊂ Fq , defined over Fq induces non-linear Hamming isometries between the finite Zariski tangent spaces to X.
1 1.1
The genericity index and the generic minimum distance of a tangent code Preliminaries on the Zariski tangent spaces to an affine variety
Let Fq = ∪∞ m=1 Fq m be the algebraic closure of the finite field Fq with q elements and n n Fq be the n-dimensional affine space over Fq . An affine variety X ⊂ Fq is the common zero set X = V (f1 , . . . , fl ) = {a ∈ Fq
n
|f1 (a) = . . . = fl (a) = 0} n
of polynomials f1 , . . . , fl ∈ Fq [x1 , . . . , xn ]. We say that X ⊂ Fq is defined over Fq n and denote it by X/Fq ⊂ Fq if the absolute ideal I(X, Fq ) := {f ∈ Fq [x1 , . . . , xn ] | f (a) = 0, ∀a ∈ X} of X is generated by polynomials g1 , . . . , gm ∈ Fq [x1 , . . . , xn ] with coefficients from n Fq . The affine subvarieties of Fq form a family of closed subsets. The corresponding n n topology is referred to as the Zariski topology on Fq . An affine variety X ⊆ Fq is irreducible if any decomposition X = Z1 ∪Z2 of X into a union of Zariski closed subsets n Zj ⊆ X has X = Z1 or X = Z2 . If X/Fq ⊂ Fq is an irreducible affine variety, defined over Fq , then for any constant field Fq ⊆ F ⊆ Fq the function field F (X) of X over F is defined as the fraction field of the affine coordinate ring F [X] := F [x1 , . . . , xn ]/I(X, F ) of X over F . The local ring Oa (X, F ) of an F -rational point a ∈ X(F ) := X ∩ F n in 1 X over F consists of the quotients ϕ ϕ2 of ϕ1 , ϕ2 ∈ F [X] with ϕ2 (a) 6= 0. An F -linear derivation Da : Oa (X, F ) → F at a ∈ X(F ) is an F -linear map, subject to LeibnitzNewton rule Da (ψ1 ψ2 ) = Da (ψ1 )ψ2 (a) + ψ1 (a)Da (ψ2 ) for derivation of a product of ψ1 , ψ2 ∈ Oa (X, F ) at a. The F -linear space Ta (X, F ) = Dera (Oa (X, F ), F ) of the F -linear derivations Da : Oa (X, F ) → F at a ∈ X(F ) is called the Zariski tangent space to X at a. The derivation rule of a product of rational functions supplies the derivation rule Da (ϕq )ϕ2 (a) − ϕ1 (a)Da (ϕ2 ) ϕ1 = Da ϕ2 ϕ2 (a)2 of a quotient of ϕ1 , ϕ2 ∈ F [X] ϕ2 6= 0. Therefore any at derivations Da : F [X] → F at a ∈ X(F ) has uniquely determined extension to a derivation Da : Oa (X, F ) → F at a and there is an F -linear isomorphism Ta (X, F ) ≃ Dera (F [X], F ). Further, any derivation Da : F [X] → F lifts to a derivation Da : F [x1 , . . . , xn ] → F , vanishing on the ideal I(X, F ) of X overF . If f1 , . . . , fm ∈ F [x1 , . . . , xn ] is a generating set of m m P P I(X, F ) ⊂ I(a, F ), then Da fi gi = Da (fi )gi (a) reveals that i=1
i=1
Ta (X, F ) ≃ {Da ∈ Dera (F [x1 , . . . , xn ], F ) | Da (f1 ) = . . . = Da (fm ) = 0}. 3
Let us view the polynomials F [x1 , . . . , xn ] = F [x1 − a1 , . . . , xn − an ] as a graded ring (i) F [x1 , . . . , xn ] = ⊕∞ i=0 F [x1 − a1 , . . . , xn − an ] ,
where F [x1 − a1 , . . . , xn − an ](i) is the F -linear space of the homogeneous polynomials of x1 − a1 , . . . , xn − an of degree i. Note that an arbitrary F -linear derivation Da : F [x1 , . . . , xn ] → F at a ∈ F n vanishes on F [x1 − a1 , . . . , xn − an ](0) = F , according to Da (1) = Da (1.1) = Da (1).1 + 1.Da (1) = 2Da (1). Due to Da |F [x1−a1 ,...,xn−an ](i) = 0 for all i ≥ 2, the derivation Da is determined uniquely by its F -linear restriction Da : F [x1 − a1 , . . . , xn − an ](1) = SpanF (x1 − a1 , . . . , xn − an ) −→ F
on the n-dimensional space F [x1 − a1 , . . . , xn − an ](1) of the homogeneous linear polynomials. That enables to view the Zariski tangent space n
Ta (Fq , F ) ≃ Dera (F [x1 , . . . , xn ], F ) n
to the n-dimensional affine space Fq as a subspace of the F -linear functionals HomF (F [x1 − a1 , . . . , xn − an ](1) , F ) on F [x1 − a1 , . . . , xn − an ](1) . Since any F linear map L : F [x1 − a1 , . . . , xn − an ](1) → F extends to an F -linear derivation F [x1 , . . . , xn ] → F at a, one has an F -linear isomorphism Dera (F [x1 , . . . , xn ], F ) ≃ HomF (F [x1 − a1 , . . . , xn − an ](1) , F ). Let us denote by ∂x∂ j , 1 ≤ j ≤ n the uniquely determined F -linear derivations ∂ : F [x1 , . . . , xn ] = F [x1 − a1 , . . . , xn − an ] −→ F ∂xj a
at a, which restrict to the dual basis of xj − aj , 1 ≤ j ≤ n on F [x1 − a1 , . . . , xn − an ](1) . These are defined by the equalities ( 1 for 1 ≤ i = j ≤ n, ∂ (xi − ai ) = δij = ∂xj a 0 for 1 ≤ i 6= j ≤ n. As a result, the Zariski tangent space to n X ∂ vj Ta (X, F ) = v = ∂xj a j=1
X at a ∈ X(F ) over F can be described as n X ∂fi v (a) = 0, ∀1 ≤ i ≤ m j ∂xj j=1
for any generating set f1 , . . . , fm of I(X, F ). Let us denote by ∂f1 ∂f1 . . . ∂xn ∂(f1 , . . . , fm ) ∂x1 = ... ... ... ∂(x1 , . . . , xn ) ∂fm m . . . ∂f ∂x1 ∂xn
the Jacobian matrix of f1 , . . . , fm with respect to x1 , . . . , xn and note that Ta (X, F ) ∂(f1 ,...,fm ) 1 ,...,fm ) n is the right null-space RN ∂(f ∂(x1 ,...,xn ) (a) of ∂(x1 ,...,xn ) (a) in F , i.e., the solution set of 1 ,...,fm ) the homogeneous linear system with matrix ∂(f ∂(x1 ,...,xn ) (a) ∈ Matm×n (F ). In terms of the coding theory, any Zariski tangent space Ta (X, F ) ⊂ F n over a finite field F is 1 ,...,fm ) the linear code of length n with parity check matrix ∂(f ∂(x1 ,...,xn ) (a).
4
1.2
Puncturing of linear codes and affine varieties n
For an arbitrary field Fq ⊆ F ⊆ Fq and arbitrary indices 1 ≤ i1 < . . . < id−1 ≤ n, let us denote by Πi : F n −→ F n , Πi (x1 , . . . , xn ) = (. . . , xc [ i1 , . . . , x id−1 , . . .)
the puncturing at i = {i1 , . . . , id−1 }. We start with a lemma, which characterizes the minimum distance of an Fq -linear code C ⊂ Fnq by its puncturings. Lemma 1. An Fq -linear code C ⊂ Fnq of length n and dimension dimFq C = k ∈ N is of minimum distance d ≤ n + 1 − k if and only if for any (d − 1)-tuple of indexes i = {i1 , . . . , id−1 }, the puncturing Πi : C → Πi (C) of C at i is an Fq -linear embedding and there is a puncturing Πj : C → Πj (C) at d coordinates j = {j1 , . . . , jd }, which is not injective. Proof. By reductio ad absurdum, suppose that C ⊂ Fnq is of minimum distance d and there is a puncturing Πi : C → Fqn−d+1 at i = {i1 , . . . , id−1 } with non-trivial kernel ker Πi |C 6= {0n }. Then any c ∈ ker Πi |C \ {0n } has support Supp(c) := {1 ≤ s ≤ n | cs 6= 0} ⊆ {i1 , . . . , id−1 } and c is of weight ≤ d − 1. That contradicts d(C) = d and justifies the injectiveness of Πi : C → Fqn−d+1 for ∀i = {i1 , . . . , id−1 }, Further, for any word c ∈ C of weight d the puncturing Πj : C → Fqn−d at its support Supp(c) = {j1 , . . . , jd } contains c 6= 0n in its kernel. Therefore Πj |C is not injective. Conversely, let us assume that for any i = {i1 , . . . , id−1 } ⊂ {1, . . . n} the puncturing Πi : C → Fqn−d+1 is injective and there exists a non-injective puncturing Πj : C → Fqn−d at some j = {j1 , . . . , jd } ⊆ {1, . . . , n}. If there is a word c ∈ C of weight 1 ≤ s ≤ d − 1, then for any {i1 , . . . , id−1 } ⊇ {α1 , . . . , αs } = Supp(c) the puncturing Πi : C → Fqn−d+1 contains c ∈ ker Πi |C \ {0n } in its kernel. That contradicts the assumption ker Πi |C = {0n } and shows that the minimum distance of C is d(C) ≥ d. On the other hand, any word a ∈ ker Πj |C \ {0n } has support Supp(a) ⊆ {j1 , . . . , jd }. The non-existence of words a of weight wt(a) ≤ d − 1 specifies that Supp(a) = {j1 , . . . , jd } and a is of weight d. Thus, d(C) = d. In order to characterize the minimum distance of a generic Zariski tangent space Ta (X, F ), a ∈ X(F ) by a global geometric property of X, let us note that the puncturing Πi : Ta (X, F ) → F n−|i| of the Zariski tangent space to X at a coincides with the differential of the same puncturing Πi : X → Πi (X) of X. We develop an analogue of the Implicit Function Theorem and its inverse statement, relating a puncturing of an affine variety with its differentials. In order to formulate precisely, let us recall the usual statement of the Implicit Function Theorem. Let f = (f1 , . . . , fm ) : Rn → Rm be an ordered m-tuple of continuously differentiable functions fi : Rn → R with 5
1 ,...,fm ) (a) 6= 0 at some point a ∈ Rn . Then there exist a Euclidean neighdet ∂(x∂(f n−m+1 ,...,xn ) borhood U ′ ⊆ Rn−m of a′ = (a1 , . . . , an−m ), a Euclidean neighborhood U ′′ ⊆ Rm of a′′ = (an−m+1 , . . . , an ) and a continuously differentiable local section g : U ′ → U ′′ of the puncturing
Π′′ = Π{n−m+1,...,n} : f −1 f (a) ∩ (U ′ × U ′′ ) = {(x′ , g(x′ )) | x′ ∈ U ′ } ≃ U ′ −→ U ′ of the local fibre of f over f (a) at the last m coordinates. Note that the assump1 ,...,fm ) tion det ∂(x∂(f (a) 6= 0 is equivalent to the existence of homogeneous linear n−m+1 ,...,xn ) ′ functions xn−m+1 (x ), . . . , xn (x′ ) of x′ = (x1 , . . . , xn−m ), parameterizing the right null-space RN
∂(f1 , . . . , fm ) (a) = {(x′ , xn−m+1 (x′ ), . . . , xn (x′ )) | x′ ∈ Rn−m } ≃ Rn−m ∂(x1 , . . . , xn )
of the Jacobian matrix of f1 , . . . , fm at a. In the case of polynomials fi ∈ R[x1 , . . . , xn ], 1 ≤ i ≤ m, the fibre f −1 f (a) ⊂ Rn ⊂ Cn consists of the R-rational points of the affine variety X := V (f1 − f1 (a), . . . , fm − fm (a)) ⊂ Cn . The Zariski tangent space Ta (, R) ⊆ RN
∂(f1 , . . . , fm ) ∂(f1 − f1 (a), . . . , fm − fm (a)) (a) = RN (a), ∂(x1 , . . . , xn ) ∂(x1 , . . . , xn )
as far as f1 − f1 (a), . . . , fm − fm (a) ∈ I(X, R). If we assume that the affine variety X ⊂ Cn is irreducible and Π′′ = (dΠ′′ )a : Ta (X, R) → Rn−m is injective, then there exists a Zariski open subset W ⊆ Cn , such that Π′′ : X ∩ W → Π′′ (X ∩ W ) is a finite morphism. Our Lemma 3 proves this statement in a form, which is convenient for the characterization of the minimum distance of a finite Zariski tangent space to an affine variety. Conversely, if Π′′ : X ∩ W → Π′′ (X ∩ W ) is a finite morphism then Π′′ is finite and unramified at a generic point b ∈ X ∩ W and its differential (dΠ′′ )b = Π′′ : Tb (X, R) → TΠ′′ (b) (Π′′ (X), R) is injective. In the case of a constant field F of prime n characteristic p = charF , one has to assume that the degree of X ⊂ F is relatively prime to p, in order to assert that the finite morphism Π′′ : X ∩ W → Π′′ (X ∩ W ) is unramified at a generic point b ∈ X ∩ W . In order to introduce the genericity index of an affine variety, let us recall that a morphism f : X → Y of affine varieties is finite if its generic fibres are finite. Equivalently, f : X → Y is finite if induces a finite extension f ∗ : Fq (f (X)) ֒→ Fq (X) of the corresponding absolute function fields. For an arbitrary irreducible k-dimensional n affine variety X ⊂ Fq and any d ∈ N with d − 1 ≤ n − k, we claim the existence of a finite puncturing Πi : X → Πi (X) at some i = {i1 , . . . , id−1 }. More precisely, if xτ1 + I(X, Fq ), . . . xτk + I(X, Fq ) ∈ Fq (X) is a coordinate transcendence basis of Fq (X) over Fq and ρ = {ρ1 , . . . , ρn−k } = {1, . . . , n} \ {τ1 , . . . , τk }, then the puncturing Πρ : X → Πρ (X) is a finite morphism. For any {i1 , . . . , id−1 } ⊆ {ρ1 , . . . , ρn−k } the puncturing Πρ factors through the puncturing Πi : X → Πi (X) at i = {i1 , . . . , id−1 } and the puncturing Πiρ\i : Πi (X) → Πρ (X) at ρ \ i := {ρ1 , . . . , ρn−k } \ {i1 , . . . , id−1 }.
6
In other words, there is a commutative diagram Πi
✲ ΠI (X)
X Πρ
❄✠
Πiρ\i
Πρ(X) with finite morphism Πρ . Therefore Πi : X → Πi (X) is finite and X admits a finite puncturing Πi : X → Πi (X) at d − 1 coordinates. n
Definition 2. An affine variety X ⊂ Fq is (d − 1)-generic if for any (d − 1)-tuple of n−d+1 at i indices i = {i1 , . . . , id−1 } ⊂ {1, . . . , n} the puncturing Πi : X → Πi (X) ⊆ Fq n−d is a finite morphism and there exists a non-finite puncturing Πj : X → Πj (X) ⊆ Fq of X at d coordinates j = {j1 , . . . , jd }. The next lemma can be viewed as an analogue of the Implicit Function Theorem. n
Lemma 3. Let X/Fq ⊂ Fq be an irreducible affine variety, defined over Fq and a ∈ X(Fq ) be an Fq -rational point of X, at which the Fq -Zariski tangent space Ta (X, Fq ) is of minimum distance d(Ta (X, Fq ) ≥ d. Then the genericity index of X is ≥ (d − 1). sing ) is a proper Zariski Proof. We claim that Z := X sing ∪ ∪i,|i|=d−1 Π−1 i (Πi (X) closed subset of X. First of all, the singular locus X sing is a proper affine subvariety of X. For any (d − 1)-tuple of indices i = {i1 , . . . , id−1 }, Πi (X)sing is a proper closed n−d+1 . The continuity of the puncturings subset of the quasi-affine variety Πi (X) ⊂ Fq sing ) are Πi : X → Πi (X) with respect to the Zariski topology implies that Π−1 i (Πi (X) closed subsets of X. There are finitely many (d−1)-tuples of indices i = {i1 , . . . , id−1 }, so that Z is a Zariski closed subset of X. Towards Z 6= X, it suffices to note that the sing ) = X implies Π (X) = Π (X)sing . As a result, the Zariski assumption Π−1 i i i (Πi (X) sing closure Πi (X) = Πi (X) Πi (X), which is an absurd. Thus, Z X is a proper Zariski closed subset of X and Wo := X \ Z is a non-empty Zariski open subset of X. According to Lemma 1, the assumption d(Ta (X, Fq )) ≥ d amounts to the injectiveness of the differentials (dΠi )a : Ta (X, Fq ) → TΠi (a) (Πi (X), Fq ) for all i = {i1 , . . . , id−1 } ⊂ {1, . . . , n}. Let us fix some generators f1 , . . . , fm ∈ I(X, Fq ) of I(X, Fq ) = hf1 , . . . , fm iFq ⊳ Fq [x1 , . . . , xn ] and recall that Ta (X, Fq ) is the Fq -linear 1 ,...,fm ) code with parity check matrix ∂(f ∂(x1 ,...,xn ) (a). The Fq -linear injectiveness of (dΠi )a is t ∂f1 ∂fm equivalent to the linear independence of the columns ∂x (a), . . . , , 1 ≤ s ≤ d−1 ∂xi i s
s
∂(f1 ,...,fm ) ∂(x1 ,...,xn ) (a),
labeled by i. This, in turn, amounts to the existence of (d − 1)-tuples of of indices r(i) = {r(i)1 , . . . , r(i)d−1 } ⊆ {1, . . . , m}, such that the minor ∆(r(i), i)a := det
∂(fr(i)1 , . . . fr(i)d−1 ) (a) 6= 0 ∂(xi1 , . . . , xid−1 )
is non-zero. Let W ⊂ X be the Zariski open neighborhood of a, which consists of the points b ∈ X with ∆(r(i), i)b 6= 0 for all i = {i1 , . . . , id−1 } ⊂ {1, . . . , n} and their 7
associated (d − 1)-tuples of indices r(i) = {r(i)1 , . . . , r(i)d−1 } ⊆ {1, . . . , m}. By the irreducibility of X, the intersection W ∩ Wo 6= ∅ is a non-empty, Zariski open, Zariski dense subset of X. Let us choose some b ∈ W ∩ Wo and denote by Fqm the definition field of b, so that b ∈ (W ∩ Wo )(Fqm ). Then the presence of Fqm -linear embeddings (dΠi )b : Tb (X, Fqm ) −→ TΠi (b) (Πi (X), Fqm ) for all i = {i1 , . . . , id−1 } implies the inequalities dim X = dimFqm Tb (X, Fqm ) ≤ dimFqm TΠi (b) (Πi (X), Fqm ) = dim Πi (X) ≤ dim X. Therefore dim X = dim Πi (X) and Πi : X → Πi (X) are finite morphisms for all i = {i1 , . . . , id−1 }. In other words, the genericity index of X is ≥ d − 1. Here is an analogue of the reversed statement of the Implicit Function Theorem. n
Lemma 4. Let X/Fq ⊂ Fq be an irreducible affine variety, defined over Fq , whose degree deg X is not divisible by the characteristic p = charFq and whose genericity index is ≥ d − 1. Then there exist a polynomial ∆ ∈ Fqmo [x1 , . . . , xn ] and a natural number m1 ∈ N, such that the minimum distance of Ta (X, Fqm ) at any point a ∈ (X \ V (∆))(Fqm ) with m ∈ m1 N is d(Ta (X, Fqm ) ≥ d. Proof. For an arbitrary (d − 1)-tuple of indices i = {i1 , . . . , id−1 }, let j = {j1 , . . . , jn−d+1 } := {1, . . . , n} \ {i1 , . . . , id−1 } be the complement of i to {1, . . . , n}. The finite morphism Πi : X → Πi (X) induces a finite extension Π∗i : Fq (Πi (X)) ֒→ Fq (X) of the corresponding (absolute) function fields. For any 1 ≤ s ≤ d − 1 let gis ,j (t) ∈ Fq (Πi (X))[t] be the minimal polynomial of xis + I(X, Fq ) ∈ Fq (X) over Fq (Πi (X)). The multiplication of gis ,j (t) by the least common multiple of the denominators of its coefficients provides a polynomial ψis ,j ∈ Fq [xis , xj ] ∩ I(X, Fq ) of minimal positive degree with respect to xis . Let ∆ :=
Y
d−1 Y
i,|i|=d−1 s=1
∂ψis ,j ∈ Fq [x1 , . . . , xn ]. ∂xis
The polynomial ∆ has finitely many coefficients and belongs to Fqmo [x1 , . . . , xn ] for some mo ∈ N. We claim that ∆ 6∈ I(X, Fq ), so that X \ V (∆) 6= ∅ is a non-empty, ∂ψ Zariski open subset of X. The assumption ∆ ∈ I(X, Fq ) implies ∂xiis ,j ∈ I(X, Fq ) s for some i = {i1 , . . . , id−1 } and some 1 ≤ s ≤ d − 1, as far as the absolute ideal n I(X, Fq ) ⊳ Fq [x1 , . . . , xn ] of the irreducible affine variety X ⊂ Fq is prime. We have chosen the polynomial ψis ,j ∈ Fq [xis , xj ] ∩ I(X, Fq ) to be of minimal degree with ∂ψ ∂ψ respect to xis , so that its partial ∂xiis ,j belongs to I(X, Fq ) if and only if ∂xiis ,j ≡ 0 s s vanishes identically. The last condition happens exactly when the exponents of xis in all the monomials of ψis ,j (with non-zero coefficients) are divisible by the characteristic p = charFq . In particular, the degree degxis ψis ,j of ψis ,j with respect to xis is to be 8
divisible by p. By the very definition of ψis ,j , the puncturing Πi : X → Πi (X) at d−1 Q i = {i1 , . . . , id−1 } is of degree deg Πi = degxis ψis ,j , so that deg Πi turns to be a s=1
multiple of p. By assumption, Πi : X → Πi (X) is a finite morphism, so that Πi (X) is of dimension dim Πi (X) = dim X = k and j = {j1 , . . . , jn−d+1 } = {1, . . . , n} \ i contains a subset τ = {τ1 , . . . , τk }, labeling a coordinate transcendence basis xτ1 + I(X, Fq ), . . . , xτk + I(X, Fq ) ∈ Fq [X] of Fq (X) over Fq . Therefore the complement ρ = {ρ1 , . . . , ρn−k } = {1, . . . , n} \ {τ1 , . . . , τk } of τ contains i and the finite dominant k puncturing Πρ : X → Fq factors through the puncturing Πi : X → Πi (X). In other words, there is a commutative diagram X
Πi
✲ Πi (X)
Πρ
Πiρ\i
❄✠
Πρ (X) of finite surjective morphisms, where Πiρ\i : Πi (X) → Πρ (X) stands for the puncturing at the complement ρ \ i of i to ρ. Thus, the degree deg X = deg Πρ |X = deg Πiρ\i |Πi (X) deg Πi |X of X is a multiple of the degree of Πi |X and, therefore, p = Fq divides deg X. That contradicts the assumptions of the lemma and justifies that X \ V (∆) 6= ∅ is a non-empty Zariski open subset of the irreducible variety X. In particular, (X \ V (∆))(Fqm1 ) 6= ∅ for a sufficiently large natural number m1 and (X \ V (∆))(Fqm ) 6= ∅ for all m ∈ m1 N. For an arbitrary i = {i1 , . . . , id−1 }, let us note that the finite covering Πi : X \ V (∆) −→ Πi (X \ V (∆)) is unramified, as far as the branch locus of Πi |X is X ∩ V
d−1 Q ∂ψis ,j s=1
∂xis
. Therefore
Πi |X\V (∆) is etale and its differential (dΠi )a : Ta (X, Fqm ) → TΠi (a) (Πi (X), Fqm ) is injective at any a ∈ (X \ V (∆))(Fqm ). By Lemma 1, the minimum distance d(Ta (X, Fqm ) ≥ d . Here is the main result of the present section n
Theorem 5. Let X/Fq ⊂ Fq be an irreducible, (d − 1)-generic affine variety, defined over Fq with GCD(deg X, charFq ) = 1. Then there exist ∆ ∈ Fqmo [x1 , . . . , xn ] \ {0} and m1 ∈ N, such that the Zariski tangent spaces Ta (X, Fqm ) to X at all the points a ∈ (X \ V (δ))(Fqm ) with m ∈ m1 N are of minimum distance d(Ta (X, Fqm ) = d. n
Proof. According to Lemma 4, for a (d − 1)-generic irreducible variety X/Fq ⊂ Fq with GCD(deg X, charFq ) = 1 there exist ∆ ∈ Fqmo [x1 , . . . , xn ] and m1 ∈ N, such that d(Ta (X, Fqm ) ≥ d for all a ∈ (X \ V (∆))(Fqm ) and all m ∈ m1 N. The assumption 9
d(Ta (X, Fqm )) ≥ d + 1 for some a ∈ (X \ V (∆))(Fqm ) requires X to be of genericity index ≥ d by Lemma 3. The contradiction justifies d(Ta (X, Fqm )) = d for ∀a ∈ (X \ V (∆))(Fqm ), ∀m ∈ m1 N.
2 2.1
Algorithm for obtaining the generic minimum distance of a tangent code and the points, at which it is attained The decomposition into irreducible components, the degree and the dimension
The input of our algorithm consists of an irreducible (d − 1)-generic affine variety n X/Fq ⊂ Fq of GCD(deg X, charFq ) = 1, given by a generating set f1 , . . . , fm ∈ Fq [x1 , . . . , xn ] of its absolute ideal I(X, Fq ) = hf1 , . . . , fm iFq ⊳ Fq [x1 , . . . , xn ]. Such a description of an irreducible affine variety can be obtained by starting with arbitrary polynomials g1 , . . . , gr ∈ Fq [x1 , . . . , xn ] and computing the primary decomposition of the ideal hg1 , . . . , gr iFq ⊳ Fq [x1 , . . . , xn ], generated by them. For an explanation and comparison of algorithms, executing primary decomposition of a polynomial ideal, one can confer the article [2] of Decker, Greuel and Pfister. Let hg1 , . . . , gr iFq = Q1 ∩ .√ . . ∩ Qt be the decomposition into primary ideals with different prime radicals √ Pi = Qi ⊳ Fp Qi , which do not q [x1 , . . . , xn ] of Qi , 1 ≤ i ≤ t. The ideals Pi = contain Pj = Qj for some j 6= i are called minimal. If P1 , . . . , Ps for some s ≤ t are the minimal radicals of the considered primary decomposition then the affine variety n Y = V (g1 , . . . , gr ) = V (P1 )∪. . . V (Ps ) ⊂ Fq decomposes into irreducible components V (Pi ) with explicitly given generators g1 , . . . , gr , gi,1 , . . . , gi,νi ∈ Fq [x1 , . . . , xn ] of PI ⊳ Fq [x1 , . . . , xn ]. There exist effective algorithms for computing the dimension k and the degree D n of an affine variety X = V (f1 , . . . , fm )/Fq ⊂ Fq . More precisely, if I = hf1 , . . . , fm iFq is the ideal, generated by the equations of X, then the Hilbert function HFX (s) = HFI (s) := dimFq Fq [x1 , . . . , xn ]≤s /I ≤s
of X is defined as the Hilbert function of the ideal I, i.e., as the dimension of the quotient space of the polynomials of total degree ≤ s with respect to the subspace I ≤s of the polynomials from I of degree ≤ s. Hilbert has shown that for a sufficiently large s ∈ N, the Hilbert function HFX (s) is a polynomial of s with leading terms D k s , k!
where k = dim X and D = deg X. That is why, it suffices to compute the Hilbert polynomial HFI (s), in order to obtain k = dim X = deg HFI (s) and D = deg X = [deg HFX (s)]!LC(HFX (s)). In order to outline an algorithm for computing HFI (s), let us recall that a monomial order > of Fq [x1 , . . . , xn ] is graded if xα > xβ for all the monomials of total degree |α| > |β| and > restricts to a lexicographic order on the monomials of equal total degree. The ideal hLT (I)i of the leading terms of I is 10
generated by the leading terms LT (f ) of all the entries f ∈ I of I with respect to the fixed monomial order. According to Proposition 4 from Chapter 9, §3 [1], the Hilbert polynomials HFI (s) = HFhLT (I)i (s) coincide. If G is a Groebner basis of I with respect to the considered monomial order, then the finite set LT (G) generates hLT (I)i. For an arbitrary monomial ideal J = hxα(1) , . . . , xα(s) i ⊳ F [x1 , . . . , xn ], let M C(J)≤s be the set of the monomials of degree ≤ s from the complement of J. Then the Hilbert polynomial HFJ (s) = |M C(J)≤s |. By Theorem 3 from Chapter α(i) 9, § 2 [1], the monomial complement M C(J) = ∪ti=1 M (xAi F [xBi ]) consists of the α(i)
α(i)
monomials M (xAi F [xBi ]) from xAi F [xBi ] for some decompositions {1, . . . , n} = Ai ∪ Bi with Ai ∩ Bi = ∅ and some α(i) ∈ (Z≥0 )|Ai | . Bearing in mind that the α(j) α(i) intersections xAi F [xBi ] ∩ xAj F [xBj ] = xαA F [xB ] are of the same form for B :=
Bi ∩ Bj , A := {1, . . . , n} \ B and an appropriate α ∈ (Z≥0 )|A| , one computes explicitly |M C(hLT (I)i≤s | = HFhLT (I)i (s) = HFI (s) = HFX (s) and obtains k = dim X, D = deg X.
2.2
The coordinate transcendence bases and the genericity index n
Let X/Fq ⊂ Fq be an irreducible k-dimensional affine variety, defined over Fq and Fq ⊆ F ⊂ Fq be a finite extension of Fq . For an arbitrary polynomial g ∈ Fq [x1 , . . . , xn ] there is a finite extension F1 ⊇ F , containing all the coefficients of g(x1 , . . . , xn ). The finite product Y go (x1 , . . . , xn ) := σg(x1 , . . . , xn ) σ∈Gal(F1 /F )
is from F [x1 , . . . , xn ], as far as the action of the absolute Galois group Gal(F = Fq /F ), on σg(x1 , . . . , xn ) ∈ F1 [x1 , . . . , xn ] reduces to the action of Gal(F1 /F ) and stabilizes go (x1 , . . . , xn ). That is why, xτ1 + I(X, Fq ), . . . , xτk + I(X, Fq ) ∈ Fq [X] = Fq [x1 , . . . , xn ]/I(X, Fq ) is a coordinate transcendence basis of Fq (X) over Fq exactly when xτ1 + I(X, F ), . . . , xτk + I(X, F ) ∈ F [X] constitute a coordinate transcendence basis of F (X) over F . A monomial xτ := xτ1 . . . xτk of degree deg xτ = k = dim X is said to be associated with a coordinate transcendence basis of X if for any constant field F , containing the definition field Fq of X, the elements xτ1 + I(X, F ), . . . , xτk + I(X, F ) ∈ F [X] ⊂ F (X) form a transcendence basis of F (X) over F . The next lemma provides a combinatorial algorithm for computing the genericity index of an affine variety. n
Lemma 6. Let X/Fq ⊂ Fq be an irreducible affine variety, whose degree deg X is not divisible by p = charFq and xτ (1) , . . . , xτ (s) be the monomials, associated with all the coordinate transcendence bases of X. If M := {J ⊆ {1, . . . , n} | J ∩ {τ (i)1 , . . . , τ (i)k } = 6 ∅
for
∀1 ≤ i ≤ s}
and d := minJ∈M |J| then dim V (xτ (1) , . . . , xτ (s) ) = n − d and X is (d − 1)-generic. Proof. The equality dim V (xτ (1) , . . . , xτ (s) ) = n − min |J| J∈M
11
is well known property of the common zero sets of monomials (e.g., cf. Proposition 3 from Chapter 9, §1 of Cox, Little and O’Shea’s [1]). In order to check that X is (d−1)-generic for d := minJ∈M |J|, let us note that for any subset i = {i1 , . . . , id−1 } ⊂ {1, . . . , n} of cardinality |i| = d − 1 there exists an index 1 ≤ j ≤ s with I ∩ {τ (j)1 , . . . , τ (j)k } = ∅. The coordinate transcendence basis xτ (j)1 , . . . , xτ (j)k of X is associated with the finite puncturing Πρ : X → Πρ (X) at ρ = {1, . . . , n} \ {τ (j)1 , . . . , τ (j)k }. The inclusion i ⊆ {1, . . . , n} \ {τ (j)1 , . . . , τ (j)k } = {ρ1 , . . . , ρn−k } implies that the finite puncturing Πρ : X → Πρ (X) factors through the puncturing Πi : X → Πi (X) at i = {i1 , . . . , id−1 }. Thus, Πi : X → Πi (X) is a finite morphism for any subset i ⊂ {1, . . . , n} of cardinality |i| = d − 1 and the genericity index of X is ≥ d − 1. Let us assume that all the puncturings of X at d = minJ∈M |J| coordinates are finite morphisms of X and choose Jo ∈ M with |Jo | = d. Since ΠJo : X → ΠJo (X) is a finite morphism, the complement α := {1, . . . , n} \ Jo contains the label set τ (i) = {τ (i)1 , . . . , τ (i)k } of a coordinate transcendence basis xτ (i)1 , . . . , xτ (i)k of X. As a result, Jo ⊆ {1, . . . , n} \ {τ (i)1 , . . . , τ (i)k } contradicts Jo ∩ {τ (i)1 , . . . , τ (i)k } = 6 ∅ and justifies that X is d − 1-generic.
2.3
The discriminant locus of the (d − 1)-puncturings
Here is an effective procedure for obtaining the polynomials ψis ,j ∈ Fq [xis , xj ] \ Fq [xj ] and, therefore, the polynomial ∆ :=
Y
d−1 Y
i,|i|=d−1 s=1
∂ψis ,j ∂xis
from Theorem 5. For any i = {i1 , . . . , id−1 } with complement j = {j1 , . . . , jn−d+1 } = {1, . . . , n} \ {i1 , . . . , id−1 } and any 1 ≤ s ≤ d − 1, let us consider the lexicographic order of the monomials of x1 , . . . , xn with xi\is > xis > xj . Suppose that I(X, Fq ) = hf1 , . . . , fm iFq ⊳ Fq [x1 , . . . , xn ] is generated by f1 , . . . , fm ∈ Fq [x1 , . . . , xn ] and compute a Groebner basis Gis ,j = {g1 , . . . , gt } of I := hf1 , . . . , fm iFq ⊳ Fq [x1 , . . . , xn ] with respect to this lexicographic order. We claim that the intersection G′′is ,j := Gis ,j ∩ {Fq [xis , xj ] \ Fq [xj ]} consists of a unique polynomial ψis ,j ∈ I(X, Fq )∩ (Fq [xis , xj ]\Fq [xj ]), which is of minimal degree with respect to xis and works for the purpose. Indeed, by the Elimination Theorem 2 from Chapter 3, § 1 of Cox, Little and O’Shea’s [1], G′is ,j := Gis ,j ∩ Fq [xis , xj1 , . . . , xjn−d+1 ] 12
is a Groebner basis of the ideal I(is , j) := I ∩ Fq [xis , xj ] ⊳ Fq [xis , xj ] with respect to the lexicographic order with xis > xj . Applying Lemma 4 from [2], one concludes that G′′is ,j := G′is ,j \ Fq [xj ] is a Groebner basis of the extension I(is , j)e := I(is , j)Fq (xj )[xis ] of I(is , j) to the polynomial ring Fq (xj )[xis ] of one variable xis with coefficients from the field Fq (xj ) of the rational functions of xj = {xj1 , . . . , xj−d+1 } with Fq coefficients. The monomial order of Fq (xj )[xis ] is the usual one with xαis > xβis for α > β, α, β ∈ Z≥0 . Any ideal of Fq (xj )[xis ] is principal, so that I e (is , j) has a unique generator ψis ,j ∈ Fq [xis , xj ] ∩ Fq (xj )[xis ] and hG′′ i = hψis ,j i. In order to justify that ψis ,j ∈ I(X, Fq ) ∩ (Fq [xis , xj ] \ Fq [xj ]) is of minimal degree ao ∈ N with respect to xis , let us assume that ϕis ,j ∈ I(X, Fq ) ∩ (Fq [xis , xj ] \ Fq [xj ]) is of minimal degree bo ∈ N with respect to xis . Then bo ≤ ao . On the other hand, ϕis ,j ∈ I(is , j) := I(X, Fq ) ∩ Fq [xis , xj ], whereas ϕis ,j ∈ I e (is , j) := I(is , j)Fq (xj )[xis ]. Therefore the leading term LT (ϕis ,j ) = xbiso xbj , b ∈ (Z≥0 )n−d+1 of ϕis ,j ∈ Fq [xis , xj ] with respect to the lexicographic order with xis > xj is divisible by the leading term LT (ψis ,j ) = xaiso xj , a ∈ (Z≥0 )n−d+1 of ψis ,j , since G′′is ,j = {ψis ,j } is a Groebner basis of I e (is , j). That implies ao ≤ bo , whereas ao = bo and the polynomial ψis ,j ∈ I(X, Fq ) ∩ (Fq [xis , xj ] \ Fq [xj ]) is of minimal degree with respect to xis . One way for obtaining m1 ∈ N with (X \ V (∆))(Fqm ) 6= ∅ for all m ∈ m1 N is by using Fried, Haran and Jarden’article [3]. They show that if k := dim X, D := deg X, δ := deg ∆, then the number Nm := |(X \ V (∆))(Fqm )| of the Fqm -rational points of X \ V (∆) satisfies the inequality 1
|Nm − q mk | ≤ (D − 1)(D − 2)q m(k− 2 ) + C(n, k, D, δ)q m(k−1)
(1)
for an appropriate constant C(n, k, D, δ), depending on n, k, D, δ. More precisely, let i h m Co (n, k, D) := 2k−1 [D(D − 1)2 + 1] + k 1 + (D − 1)(D − 2) + 22n+k−3 2m m2 D 2 with m := bound
n+D k n
be Litz’s constant from [5], which works out for the Lang-Weil’s 1
′ |Nm − q mk | ≤ (D − 1)(D − 2)q m(k− 2 ) + Co (n, k, D)q m(k−1)
′ := |Y (F m )| of the F m -rational points of a smooth irreducible on the number Nm q q projective variety Y /Fq ⊂ Pn (Fq ) of dim Y = k and deg Y = D (cf. [4] of Lang and Weil). Then Fried-Haran-Jarden’s constant equals
C(n, k, D, δ) := Co (n, k, D) + 2k−1 D(δ + 1). Note that (1) implies the inequality 1
Nm ≥ q mk − (D − 1)(D − 2)q m(k− 2 ) − C(n, k, D, δ)q m(k−1) = i h m = q m(k−1) q m − (D − 1)(D − 2)q 2 − C(n, k, D, δ) . 13
In order to obtain m1 ∈ N with Nm ≥ 1 for ∀m = m1 n ∈ m1 N, it suffices to have q m1 n − (D − 1)(D − 2)q
m1 n 2
− C(n, k, D, δ) > 0
for all n ∈ N. The roots of the quadratic equation t2 − (D − 1)(D − 2)t − C = 0 with C := C(n, k, D, δ) are p √ (D − 1)(D − 2) ± (D − 1)2 (D − 2)2 + 4C C + C 2 + 4C < , 2 2 due to C > (D − 1)(D − 2). That is why, one can take √ √ m1 n C + C 2 + 4C C + C 2 + 4C + 4 2 q > . ≥ C +1 = 2 2 Thus, m1 ≥ 2 logq (C + 1) provides Nm = Nm1 n ≥ 1 for all n ∈ N. We propose an algorithm for obtaining points a ∈ X \ V (∆) by consecutive puncturings of the coordinates. Let f1 , . . . , fm , h1 , . . . , hl ∈ Fq [x1 , . . . , xn ] be such polyn nomials that X = V (f1 , . . . , fm ) Fq is an irreducible affine variety with absolute ideal I(X, Fq ) = hf1 , . . . , fm iFq and Wn := V (f1 , . . . , fm ) \ V (h1 , . . . , hl ) 6= ∅ is a nonempty Zariski open subset of X. In the case of n = 1, such X is a point. In general, there exists 1 ≤ i ≤ n, such that the puncturing Πi : X → Πi (X) at xi is a finite morphism. Denote x′ := {x1 , . . . , xi−1 , xi+1 , . . . xn } and fix a lexicographic order with xi > x′ . Obtain a Groebner basis G′ = {g1 , . . . , gs } of I = hf1 , . . . , fm iFq = I(X, Fq ) and extend to a Groebner basis G′′ of J = hf1 , . . . , fm , h1 , . . . , hl iFq . Let LCxi (G′ ) = {LCxi (g1 ), . . . , LCxi (gs )} ⊂ Fq [x′ ] be the set of the leading coefficients of the entries of G′ , viewed as polynomials of xi with coefficients from Fq [x′ ]. The Extension Theorem 3 from Chapter 3, §1 [1] asserts that for an arbitrary point a′ ∈ Vn−1 (G′ ∩ Fq [x′ ]) \ Vn−1 (LCxi (G′ )) ⊆ Fq
n−1
the polynomial system of equations g(xi , a′ ) = 0, ∀g ∈ G′ \Fq [x′ ] has a solution ai ∈ Fq and for any solution ai of this system the point a = (ai , a′ ) belongs to X. We claim that Wn−1 := Vn−1 (G′ ∩ Fq [x′ ]) \ {Vn−1 (G′′ ∩ Fq [x′ ]) ∪ Vn−1 (LCxi (G′ ))} ⊆ Fq n−1
n−1
, and for any point a′ ∈ Wn−1 the is a non-empty quasi-affine subvariety of Fq ′ ′ solution ai ∈ Fq of g(ai , a ) = 0, ∀g ∈ G \ Fq [x′ ] provides a point a = (ai , a′ ) ∈ Wn := X \ V (J). Indeed, since G′ generates I, the point a ∈ X = V (I). The assumption a ∈ V (J) implies that a′ = Πi (a) ∈ Πi V (J). By the Closure Theorem 3 from Chapter 3, §2 [1], the Zariski closure of Πi V (J) is Πi V (J) = Vn−1 (G′′ ∩ Fq [x′ ]), so that a′ ∈ Vn−1 (G′′ ∩ Fq [x′ ]), contrary to its choice. That justifies a ∈ V (I) \ V (J). n Note that X = V (I) ⊂ Fq is an irreducible affine variety. Therefore, its image n−1 under the puncturing Πi is irreducible, as well as the closure Πi (X) ⊆ Fq Πi (X) = Vn−1 (G′ ∩ Fq [x′ ]) = Wn−1 . 14
That justifies the irreducibility of Wn−1 . As a result, if ′ Wn−1 := Vn−1 (G′ ∩ Fq [x′ ]) \ Vn−1 (G′′ ∩ Fq [x′ ]), ′′ Wn−1 := Vn−1 (G′ ∩ Fq [x′ ]) \ Vn−1 (LCxi (G′ )),
′ ′′ then it suffices to check that Wn−1 6= ∅ and Wn−1 6= ∅, in order to assert that Wn−1 = ′ ′′ ′ Wn−1 ∩Wn−1 6= ∅. The assumption Wn−1 = ∅ implies the inclusion Vn−1 (G′ ∩Fq [x′ ]) ⊆ Vn−1 (G′′ ∩Fq [x′ ]). Making use of G′ ⊂ G′′ , one concludes that G′ ∩Fq [x′ ] ⊆ G′′ ∩Fq [x′ ], whereas Vn−1 (G′′ ∩ Fq [x′ ]) ⊆ Vn−1 (G′ ∩ Fq [x′ ]). Thus,
Πi (X) = Vn−1 (G′ ∩ FQ [x′ ]) = Vn−1 (G′′ ∩ Fq [x′ ]) = Πi V (J). As far as Πi : Wn → Πi (Wn ) is a finite morphism and the Zariski closure Wn = X, one has dim X = dim V (I) = dim Πi V (I) = Πi V (J) ≤ dim V (J). On the other hand, Wn := X \ V (J) 6= ∅ implies that dim V (J) < dim V (I) = dim X, which is an absurd, ′′ ′ = ∅ requires 6= ∅. In a similar vein, note that the assumption Wn−1 justifying Wn−1 Πi (X) = Vn−1 (G′ ∩ Fq [x′ ]) ⊆ Vn−1 (LCxi (G′ )), whereas LCxi (G′ ) ⊂ I(Πi (X), Fq ) = I(Πi (X), Fq ) The choice of a finite morphism Πi : X → Πi (X) forces G′ \ Fq [x′ ] 6= ∅. By Lemma 4 from [2], G′ \ Fq [x′ ] is a Groebner basis of the extension I e := IFq (x′ )[xi ] ⊳ Fq (x′ )[xi ] with respect to the induced lexicographic order of x′ . Note that Fq (x′ )[xi ] is a principal ideal domain and choose a generator g of I e = hG′ \Fq [x′ ]i⊳Fq (x′ )[xi ]. The polynomial g is a multiple of the minimal polynomial pi (t) ∈ Fq (Πi (X))[t] of xi +I(X, Fq ) ∈ Fq (X) over Fq (Πi (X)). Therefore, its leading coefficient LCxi (g) ∈ Fq [Πi (X)] \ {0} does not belong to the ′′ ideal I(Πi (X), Fq ) of Πi (X). The contradiction justifies Wn−1 6= ∅ and Wn−1 6= ∅. Finally, let P := (G′′ ∩ Fq [x′ ]).LCxi (G′ ) := {gh | g ∈ G′′ ∩ Fq [x′ ], h ∈ LCxi (G′ )} be the set of the products of the elements of G′′ ∩ Fq [x′ ] with the entries of LCxi (G′ ). Then Vn−1 (G′′ ∩ Fq [x′ ]) ∪ Vn−1 (LCxi (G′ )) = Vn−1 (P) and Wn−1 = Vn−1 (G′ ∩ Fq [x′ ]) \ Vn−1 (P).
2.4
A pseudo-code for obtaining tangent codes of generic minimum distance
The aforementioned algorithms are synthesized by the following pseudo-code. Algorithm for obtaining the generic minimum distance d of a tangent code and points a ∈ X(Fqm ) with d(Ta (X, Fqm )) = d Input: Irreducible X/Fq of dim X = k, deg X = D, GCD(D, charFq ) = 1 and f1 , . . . , fm ∈ Fq [x1 , . . . , xn ] with I(X, Fq ) = hf1 , . . . , fm iFq Output: d ∈ N, ∆ ∈ Fqmo [x1 , . . . , xn ], a ∈ X \V (∆)(Fqm ) with d(Ta (X, Fqm )) = d Step 1 - obtaining all the coordinate transcendence bases xτ of X B := ∅ 15
S := {xτ ⊂ x | |xτ | = k} FOR EACH x′′ ∈ S DO x′ := x \ x′′ G := Groebner Basis of I = hf1 , . . . , fm iFq /x′ >lex x′′ IF G ∩ Fq [x′′ ] = ∅ THEN B := B ∪ {x′′ } ELSE DO NOTHING S := S \ {x′′ } Step 2 - obtaining d ∈ N, such that X is (d − 1)-generic M := {J ⊂ {1, . . . , n} | J ∩ xτ 6= ∅ for ∀xτ ∈ B} d := min{|J| | J ∈ M} Q ∂ψis ,j Q d−1 Step 3 - obtaining ∆ := ∂xi i,|i|=d−1 s=1
s
R := {i ⊂ {1, . . . , n} | |i| = d − 1} FOR EACH i ∈ R DO j := {1, . . . , n} \ i FOR EACH 1 ≤ s ≤ d − 1 DO Gis ,j := Groebner Basis I/xi\is >lex xis >lex xj ψis ,j := Gis ,j ∩ (Fq [xis , xj ] \ Fq [xj ]) Step 4 - obtaining a ∈ Wn := X \ V (∆) Sn := {x1 , . . . , xn } In := I(X, Fq ) = hf1 , . . . , fm i Jn := hf1 , . . . , fm , ∆i Qn := f1 , . . . , fm Pn := f1 , . . . , fm , ∆ Wn := V (Qn ) \ V (Pn ) N := n M := max(dim X, 1) WHILE N > M DO CHOOSE xτ ∈ B ∩ SN CHOOSE iN ∈ {1, . . . , n} \ {τ1 , . . . , τk } TN := SN \ {xiN } G′N := Groebner Basis of IN /xiN >lex TN G′′N := Groebner Basis of JN /xiN >lex TN , G′N ⊂ G′′N IN −1 := hG′N ∩ Fq [TN ]i JN −1 := hG′′N ∩ Fq [TN ]i QN −1 := G′N ∩ Fq [TN ] PN −1 := (G′′N ∩ Fq [TN ]).LCxiN (G′N ) WN −1 := V (QN −1 ) \ V (PN −1 ) N := N − 1 M CHOOSE aM ∈ WM ⊂ Fq L := M WHILE L ≤ n DO L := L + 1 aiL := SOLUTION OF g(xiL , TL ) = 0 FOR ALL g ∈ IL \Fq [TL ]
16
3
Stabilization and destabilization of the minimum distance within a tangent bundle
The present section provides a series of k-dimensional irreducible affine varieties n X/Fq ⊂ Fq with GCD(deg X, charFq ) = 1, which have an a priori given Fq -linear [n, k, d]-code C ⊂ Fnq as its Zariski tangent space T0n (X, Fq ) = C at the origin 0n ∈ X and reproduce the length n, the dimension k and the minimum distance d of C by the Zariski tangent spaces Ta (X, Fqm ) at its generic points a ∈ X(Fqm ), m ∈ N. On the other hand, for any family C of Fq -linear codes of fixed length n, fixed dimension k and n arbitrary minimum distance, it constructs a k-dimensional affine variety X/Fq ⊂ Fq , whose degree deg X is divisible by the characteristic charFq and whose Fq -Zariski tangent bundle contains C. These statements illustrate the stabilization (respectively, the destabilization) of the minimum distance within a Zariski tangent bundle to an affine variety X, whose degree deg X is not divisible (respectively, is divisible) by the characteristic charFq .
3.1
Reproducing the minimum distance of a linear code by a Zariski tangent bundle
Proposition 7. Let C ⊂ Fnq be an Fq -linear code of length n, dimension k and minimum distance d over a field Fq of characteristic p = charFq . Then there exist n a k-dimensional affine variety X/Fq ⊂ Fq with GCD(deg X, p) = 1 and a nonempty Zariski open subset W ⊆ X smooth , such that 0n ∈ W , T0n (X, Fq ) = C and Ta (X, Fqm ) ⊂ Fnqm are Fqm -linear codes of length n, dimension k and minimum distance d for all m ∈ N and all the Fqm -rational points a ∈ W (Fqm ) of W . Proof. Let H ∈ Mat(n−k)×n (Fq ) be a parity check matrix of the k-dimensional linear code C ⊂ Fnq . Then H is of rank rk(H) = n − k and there exist linearly independent columns Hi1 , . . . , Hin−k ∈ Mat(n−k)×1 (Fq ) of H. Choose a permutation σ ∈ Sym(n) with σ(is ) = s for ∀1 ≤ s ≤ n − k and apply it to the coordinate functions x1 , . . . , xn on Fnq . That transforms the parity check matrix of C in the form H = (In−k |H ′ ), where In−k stands for the identity matrix of size n − k and H ′ ∈ Mat(n−k)×k (Fq ). Denote by H1 , . . . , Hn ∈ Mat(n−k)×1 (Fq ) the columns of H. The code C of minimum distance d contains a word c ∈ C of weight d. If Supp(c) = {j1 , . . . , jd } for some d P 1 ≤ j1 < . . . < jd ≤ n then cjs Hjs = 0(n−k)×1 with cis ∈ F∗q and s=1
Hjd = −
d−1 X
cjs c−1 jd Hjs .
s=1
Note that jd > n−k, as far as the first n−k columns of H are Fq -linearly independent. For any 1 ≤ i ≤ n − k and i ≤ j ≤ n, j 6= jd , let us denote by Hij the entry of H from the i-th row and j-th column. Then choose a polynomial X fi,j (xj ) := Hij xj + ai,j,r xrj ∈ Fq [xj ] r≥2,r6≡0(mod p)
17
and note that
∂fi,j ∂xj |xj =0
= Hij . Put
fi,jd (xjd ) := −
d−1 X s=1
cjs c−1 jd fi,js (xjd ) ∈ Fq [xjd ] for ∀1 ≤ i ≤ n − k
and observe that d−1 d−1 X X ∂fi,jd ∂fi,js (xjd ) cjs c−1 cjs c−1 =− = − jd jd Hijs = Hijd . ∂xjd xjd =0 ∂xjd xjd =0 s=1
s=1
The polynomials
fi (xi , . . . , xn ) :=
n X j=i
fi,j (xj ) ∈ Fq [xi , . . . , xn ] for 1 ≤ i ≤ n − k
are claimed to cut a k-dimensional affine variety n
X := V (f1 (x1 , . . . , xn ), f2 (x2 , . . . , xn ), . . . , fn−k (xn−k , . . . , xn ) ⊂ Fq , defined over Fq , whose degree is not divisible by p = charFq . To this end, by an induction on 0 ≤ t ≤ n − k − 1, one observes that Xt := V (fn−k−t(xn−k−t , . . . , xn ), . . . , fn−k (xn−k , . . . , xn )) ⊂ Fq
k+t+1
k
is a finite covering of the affine space Fq with coordinate functions xn−k+1 , . . . , xn . k+1 For t = 0 the hypersurface X0 := V (fn−k (xn−k , . . . , xn )) ⊂ Fq with coordinate functions xn−k , . . . , xn is k-dimensional, as far as the polynomial
= xn−k +
X
r≥2,r6≡0(mod p)
an−k,n−k,r xrn−k
fn−k (xn−k , . . . , xn ) +
n X
fn−k,j (xj )
j=n−k+1 k
depends on xn−k and the puncturing Πn−k : X0 → Πn−k (X0 ) ⊂ Fq at xn−k is a finite k morphism. If the puncturing Π(n−k−t,...,n−k) : Xt → Fq is a finite covering for some 0 ≤ t ≤ n − k − 2, then the puncturing Π(n−k−t−1) : Xt+1 → Xt is a finite covering, as far as the polynomial
= xn−k−t−1 + +
fn−k−t−1 (xn−k−t−1 , xn−k−t , . . . , xn−k ) X an−k−t−1,n−k−t−1,r xrn−k−t−1
r≥2,r6≡0(mod p) n X
j=n−k−t
fn−k−t−1,j (xj ) ∈ Fq [xn−k−t−1 , . . . , xn ]
18
depends on xn−k−t−1 and the polynomials fn−k−t(xn−k−t , xn ), . . . , fn−k (xn−k , . . . , xn ) k do not depend on xn−k−t−1 . Thus, Xn−k−1 := X is a finite covering of Fq and n X/Fq ⊂ Fq is a k-dimensional affine variety, defined over Fq . All monomials of fi (xi , . . . , xn ), 1 ≤ i ≤ n − k (with non-zero coefficients) are of degree, relatively prime to the characteristic p = charFq , so that p does not divide deg X. We claim that the absolute ideal I(X, Fq ) of X = V (f1 , . . . , fn−k ) is generated by the defining equations f1 , . . . , fn−k ∈ Fq [x1 , . . . , xn ] of X. To this end, note that f1 (x1 , . . . , xn ), . . . , fn−k (xn−k , . . . , xn ) is a Groebner basis of I := hf1 , . . . , fn−k iFq ⊳ Fq [x1 , . . . , xn ] with respect to the lexicographic order with x1 > x2 > . . . > xn . According to Theorem 3 and Proposition 4 from Chapter 2, § 9 [1], it suffices to observe that the leading monomials LM (fi ) ∈ Fq [xi ] of fi (xi , . . . , xn ) with 1 ≤ i ≤ n are pairwise relatively prime. Due to {f1 , . . . , fn−k } ∩ Fq [xn−k+1 , . . . , xn ] = ∅, the extension I e := IFq (xn−k+1 , . . . , xn )[x1 , . . . , xn−k ] ⊳ Fq (xn−k+1 , . . . , xn )[x1 , . . . , xn−k ] is a proper ideal of Fq (xn−k+1 , . . . , xn )[x1 , . . . , xn−k ]. Lemma 4 from [2] implies that f1 (x1 , . . . , xn ), . . . , fn−k (xn−k , . . . , xn ) is a Groebner basis of I e with respect to the lexicographic order with x1 > . . . > xn−k . By the very definition of the polynomials n P fi (xi , . . . , xn ) = fij (xj ) with fii (xi ) ∈ Fq [xi ]\{0}, the leading coefficients LC(fi ) ∈ j=i
∗
Fq [xn−k+1 , . . . , xn ] of fi , viewed as polynomials of x1 , . . . , xn−k belong to Fq . The Noether quotient (I : 1∞ ) := {g ∈ Fq [x1 , . . . , xn ] g1m ∈ I for some m ∈ N} = I. √ Therefore, the radical I of I is √ p I(X, Fq ) = I = (I : 1∞ ) ∩ (I, 1) = I ∩ Fq [x1 , . . . , xn ] = I
and the polynomials f1 , . . . , fn−k ∈ Fq [x1 , . . . , xn ] generate I(X, Fq ). Now, we are ready to define the non-empty Zariski open subset W ⊆ X from the statement of the proposition. By the very definition of fi (xi , . . . , xn ), 1 ≤ i ≤ n − k, the Jacobian matrix f (x ) f (x ) f1,n−k (xn−k ) f1,n (xn ) 1,2 2 1,1 1 . . . . . . ∂x2 ∂xn−k ∂xn ∂x1 ∂(f1 , . . . , fn−k ) f2,n−k (xn−k ) f2,2 (x2 ) f2,n (xn ) . = 0 ... ... ∂x2 ∂xn−k ∂xn ∂(x1 , . . . , xn ) ... ... ... ... ... ... fn−k,n (xn ) fn−k,n−k (xn−k ) ... 0 0 ... ∂xn−k ∂xn
Due to I(X, Fq ) = I = hf1 , . . . , fn−k iFq ⊳Fq [x1 , . . . , xn ], for any m ∈ N and a ∈ X(Fqm ) ∂(f ,...,f
)
n−k (a). the Zariski tangent space Ta (X, Fqm ) ⊂ Fnqm has a parity check matrix ∂(x1 1 ,...,x n) n n According to fi (0 ) = 0 for ∀1 ≤ i ≤ n, the origin 0 ∈ X = V (f1 , . . . , fn−k ) belongs to X. Moreover, ∂(f1 , . . . , fn−k ) n (0 ) = H = (In−k |H ′ ), ∂(x1 , . . . , xn )
19
so that the Zariski tangent space T0n (X, Fq ) = C. A sufficient condition for Ta (X, Fqm ) to be of dimension k over Fqm is det
fn−k,n−k f1,1 f2,2 ∂(f1 , . . . , fn−k ) (a) (an−k ) 6= 0. (a1 ) (a2 ) . . . ∂(x1 , . . . , xn ) ∂x1 ∂x2 ∂xn−k
Since C is of minimum distance d, for any i = {i1 , . . . , id−1 } ⊂ {1, . . . , n} the columns Hi1 , . . . , Hi(d−1) ∈ Mat(n−k)×1 (Fq ) of H are Fq -linearly independent. Then there exists a subset r(i) = {r(i)1 , . . . , r(i)d−1 } ⊆ {1, . . . , n − k}, such that the matrix H(r(i), i) ∈ Mat(d−1)×(d−1) (Fq ), cut by the rows of H, labeled by r(i) and the columns of H, labeled by i has det H(r(i), i) 6= 0. We define W to be the subset of X, which consists ∂(f ,...,fn−k ) (a) 6= 0 and of the points a ∈ X with det ∂(x1 1 ,...,x n) Jac(r(i), i)(a) := det
∂(fr(i)1 , . . . , fr(i)d−1 ) ∂(f1 , . . . , fn−k ) (r(i), i)(a) = det (a) 6= 0. ∂(x1 , . . . , xn ) ∂(xi1 , . . . , xid−1 )
According to 0n ∈ W , the Zariski open subset W ⊆ X is non-empty. Further, ∂(f ,...,fn−k ) dimFqm Ta (X, Fqm ) = k for all a ∈ W by det ∂(x1 1 ,...,x (a) 6= 0. We claim that n) the minimum distance d(Ta (X, Fqm )) = d for all a ∈ W . To this end, note that at any n ∂(f ,...,fn−k ) (b) is the same Fq -linear combination point b ∈ Fq , the jd -th column of ∂(x1 1 ,...,x n) of the columns, labeled by j1 , . . . , jd−1 as the jd -th column of H. Thus, the word c = (0, . . . , 0, cj1 , 0, . . . , 0, cjd , . . . , 0) ∈ C of weight d belongs to all the Fqm -linear ∂(f ,...,fn−k ) codes with parity check matrix ∂(x1 1 ,...,x (b) for some b ∈ Fnqm and, in particular, n) c ∈ Ta (X, Fqm ) for all a ∈ X(Fqm ). That justifies the upper bound d(Ta (X, Fqn ) ≤ d on the minimum distance of a tangent code at a ∈ X(Fq ). For any a ∈ W and any ∂(f ,...,f ) i = {i1 , . . . , id−1 } one has Jac(r(i), i)(a) 6= 0, so that rk ∂(xi1 ,...,xn−k ) (a) = d − 1 and i 1
d−1
Fqn−d+1 m
is injective. As a result, the minimum the puncturing Πi : Ta (X, F ) → distance d(Ta (X, Fqm )) > d − 1, whereas d(Ta (X, Fqm )) = d for ∀a ∈ W (Fqm ). qm
n
Remark: In order to have an irreducible affine variety X/Fq ⊂ Fq , subject to the properties, stated in Proposition 7, it suffices to choose fi,i (xi ) := Hii xi = 1 for ∀1 ≤ i ≤ n − k. Then for any 1 ≤ i ≤ n − k there exist polynomials gi (xi+1 , . . . , xn ), such that fi (xi , . . . , xn ) = xi − gi (xi+1 , . . . , xn ). As a result, the puncturing Π = Π(1,...,n−k) : k
X → Fq is invertible by a morphism Π−1 .
3.2
Tangent bundle interpolation of a deformation of the minimum distance with fixed length and dimension
The next proposition illustrates the possibility for incorporating codes with various minimal distances within a single Zariski tangent bundle.
20
Proposition 8. Let C → S be a family of Fq -linear codes C(a) ⊂ Fnq of length n, dimension k = dimFq C(a) and arbitrary minimum distance, parameterized by a subset n S ⊆ Fnq . Then there is a k-dimensional affine variety X/Fq ⊂ Fq , whose degree deg X is divisible by the characteristic p = charFq , such that Fnq ⊂ X, S ⊆ X smooth (Fq ) and the Zariski tangent spaces Ta (X, Fq ) = C(a) to X at a ∈ S over Fq coincide with the members of the family. Proof. Let us choose a family H → S of parity-check matrices H(a) ∈ Mat(n−k)×n (Fq ) of C(a) ⊂ Fnq for all a ∈ S and denote by H(a)ij ∈ Fq the entries of these matrices. For an arbitrary β ∈ Fq , consider the Lagrange basis polynomial LβFq (t) :=
Y
α∈Fq \{β}
t−α β−α
with LβFq (t)(β) = 1 and LβFq (t)|Fq \{β} = 0. Straightforwardly,
L0Fq (t) :=
−1 Y (t − α) (t − α) = (tq−1 − 1)(−1)−1 = −(tq−1 − 1). t=0 ∗ ∗
Y
α∈Fq
α∈Fq
Towards an explicit calculation of LβFq (t) for β ∈ F∗q , let us denote by σ1 , . . . , σq−1 Q (t − α) = tq−1 − 1. the elementary symmetric polynomials of the roots of f (t) := α∈F∗q
Put τ1 , . . . , τq−2 for the elementary symmetric polynomials of the roots of the monic polynomial fβ (t) :=
Y
α∈F∗q \{β}
q−3 X tq−1 − 1 f (t) q−2 (−1)q−2−s τq−2−s ts . = =t + (t − α) = t−β t−β s=0
Then the relations τ1 + β = σ1 = 0, τs + βτs−1 = σs = 0 for ∀2 ≤ s ≤ q − 2 and
βτq−2 = σq−1 = (−1)q ,
reveal that τs = (−β)τs−1 for ∀1 ≤ s ≤ q − 2, τ0 := 1 form a geometric progression q−2 {τs }s=1 with quotient (−β). As a result, τs = (−β)s and fβ (t) = tq−2 +
q−3 X
for ∀1 ≤ s ≤ q − 2
β q−2−s ts = tq−2 +
q−3 X s=0
s=0
21
β −s−1 ts ,
according to β q−2 = β −1 for ∀β ∈ F∗q . Now, LβFq (t)
−1
= (q − 1)
"
t
q−1
+
q−2 X
tfβ (t) := = βfβ (β)
β
−s s
t
s=1
#
"
=− t
tq−1 +
q−2 P
β −s ts
s=1 q−2 P
β q−1 +
1
s=1
q−1
+
q−2 X
β
−s s
t
s=1
#
for arbitrary β ∈ F∗q . Let us denote by n
n
Φp : Fq −→ Fq ,
Φp (a1 , . . . , an ) = (ap1 , . . . , apn ) for ∀a = (a1 , . . . , an ) ∈ Fq
n
the Frobenius automorphism of degree p = charFq and consider the polynomials
:=
X
b∈Φp (S)
n X j=1
fi (x1 , . . . , xn )
q b1 p bn p H(Φ−1 p (b))ij (xj − xj ) LFq (x1 ) . . . LFq (xn ) ∈ Fq [x1 , . . . , xn ]
n
for 1 ≤ i ≤ n − k. We claim that the affine variety X := V (f1 , . . . , fn−k ) ⊂ Fq satisfies the required consitions. First of all, Fnq ⊂ X, as far as an arbitrary point a = (a1 , . . . , an ) ∈ Fnq has components aj = aqj and fi (a1 , . . . , an ) = 0 for ∀1 ≤ i ≤ n − k. By the very definition, fi (x1 , . . . , xn ) are of degree deg fi = pn(q − 1) + q, divisible by the characteristic p = charFq , so that the degree deg X of X is a multiple of p. Straightforwardly, the partial derivatives ∂fi = ∂xj
X
b∈Φp (S)
p b1 bn p H(Φ−1 p (b))ij LFq (x1 ) . . . LFq (xn )
and their values at a ∈ S equal ∂fi (a) = H(Φ−1 p Φp (a))ij = H(a)ij . ∂xj Note that the composition of Lagrange interpolation polynomials with the Frobenius ∂(f ,...,fn−k ) (a) = H(a) at automorphism Φp is designed in such a way that to adjust ∂(x1 1 ,...,x n) all the points a ∈ S. At an arbitrary point a ∈ S ⊂ X(Fq ) := X ∩ Fnq , the Zariski tangent space Ta (X, Fq ) is contained in the right null-space of the Jacobian matrix ∂(f1 ,...,fn−k ) ∂(x1 ,...,xn ) (a) = H(a), so that Ta (X, Fq ) ⊆ C(a). Now k ≤ dim X ≤ dimFq Ta (X, Fq ) ≤ dimFq C(a) = k implies that C(a) = Ta (X, Fq ), dim X = k and any a ∈ S is a smooth point of X. 22
4
Families of Hamming isometries
A map I : Fnq → Fnq is called a Hamming isometry if it preserves the Hamming distance, i.e., d(Ix, Iy) = d(x, y) for ∀x, y ∈ Fnq , where d(x, y) := |{1 ≤ i ≤ n | xi 6= yi }| stands for the number of the different components of x and y. All Hamming isometries are bijective maps. More precisely, if we assume the existence of different x, y ∈ Fnq with Ix = Iy then 0 = d(Ix, Iy) = d(x, y) contradicts x 6= y. The present section provides a pattern for a construction of a global morphism n n n ψ : Fq → Fq and a hypersurface V (ψo ) ⊂ Fq , depending explicitly on ψ, such that the differentials of ψ restrict to linear Hamming isometries (dψ)a : Ta (X, Fqm ) −→ Tψ(a) (ψ(X), Fqm ) on the tangent codes to the affine varieties X * V (ψo ) at generic points a ∈ X \V (ψo ). Further, an arbitrary family I → S of Fq -linear Hamming isometries I(a) : Fnq → Fnq , parameterized by a subset S ⊆ Fnq is realized by the differentials (dϕ)a = I(a) of an n n appropriate Fq -morphism ϕ : Fq → Fq at a ∈ S.
4.1
Finite morphisms with isometric differentials
Proposition 9. For arbitrary polynomials ψ1 , . . . , ψn ∈ Fq [x1 , . . . , xn ] and an arbitrary permutation σ ∈ Sym(n), let us consider the morphism n
ψ := (xσ(1) ψσ(1) (xp1 , . . . , xpn ), . . . , xσ(n) ψσ(n) (xp1 , . . . , xpn )) : Fq −→ Fq
n
n
and the hypersurface V (ψo ) ⊂ Fq with equation ψo (x1 , . . . , xn ) := ψ1 (xp1 , . . . , xpn ) . . . ψn (xp1 , . . . , xpn ), where p = char(Fq stands for the characteristic of the basic field Fq . Then any irn reducible affine variety X ⊂ Fq , which is not entirely contained in the hypersurface V (ψo ) has a non-empty Zariski open, Zariski dense subset h i W := X smooth ∩ ψ −1 (ψ(X)smooth ) \ V (ψo ), such that the differentials of ψ restrict to Fqm -linear Hamming isometries (dψ)a : Ta (X, Fqm ) −→ Tψ(a) (ψ(X), Fqm ) at all the points a ∈ W (Fqm ) of W . Proof. It suffices to prove the proposition for the Fq -morphism ϕ := σ −1 ψ = (ϕ1 = x1 ψ1 (xp1 , . . . , xpn ), . . . , ϕn = xn ψn (xp1 , . . . , xpn )) : X −→ ψ(X), 23
as far as the permutation σ −1 ∈ Sym(n) coincides with its differentials at any point n a ∈ Fq and is a linear Hamming isometry. If Φ∗p : Fq [x1 , . . . , xn ] −→ Fq [x1 , . . . , xn ],
Φ∗p (f (x1 , . . . , xn )) := f (xp1 , . . . , xpn )
then the matrix of n
n
(dϕ)a : Ta (Fq , Fqm ) −→ Tϕ(a) (Fq , Fqm ) n with respect to the basis ∂x∂ j , 1 ≤ i ≤ n of Ta (Fq , Fqm ) and the basis ∂y∂ j a
n
ϕ(a)
1 ≤ j ≤ n of Tϕ(a) (Fq , Fqm ) is the Jacobian matrix ∗ Φp (ψ1 )(a) 0 ∗ (ψ )(a) ∂(ϕ1 , . . . , ϕn ) 0 Φ p 2 (a) = ... ... ∂(x1 , . . . , xn ) 0 0
,
... 0 ... 0 ... ... ∗ . . . Φp (ψn )(a)
of the components of ϕ at a. Note that at any point a ∈ (X \ V (ψo ))(Fqm ) the difn n ferential (dϕ)a : Ta (Fq , Fqm ) → Tϕ(a) (Fq , Fqm ) is an Fqm -linear Hamming isometry and restricts to an Fqm -linear Hamming isometry (dϕ)a : Ta (X, Fqm ) −→ (dϕ)a Ta (X, Fqm ) ⊆ Tϕ(a) (ϕ(X), Fqm ) onto its image. We claim that W 6= ∅ is a non-empty Zariski open subset. Due to the irreducibility of X it suffices to note that X smooth 6= ∅, X \ V (ψo ) 6= ∅ and to justify that X ∩ ϕ−1 (ϕ(X)smooth ) 6= ∅. Indeed, the assumption X ∩ ϕ−1 (ϕ(X)smooth ) = ∅ implies that ϕ(X) = ϕ(X)sing , whereas ϕ(X)smooth = ∅, which is an absurd.
4.2
Interpolation of linear Hamming isometries by differentials of a morphism
Proposition 10. Let I → S be a family of Fq -linear Hamming isometries I(a) ∈ GL(n, Fq ), I(a) : Fnq → Fnq , parameterized by a subset S ⊆ Fqn . Then there exists n n an Fq -morphism ϕ = (ϕ1 , . . . , ϕn ) : Fq → Fq , whose differentials (dϕ)a = I(a) at ∀a ∈ S coincide with the given isometries. Proof. Let us consider the polynomials n X X p q b1 bn p I(Φ−1 ϕi (x1 , . . . , xn ) := p (b))ij (xj − xj ) LFq (x1 ) . . . LFq (xn ) b∈Φp (S)
j=1
for 1 ≤ i ≤ n, where
n
n
Φp : Fq −→ Fq ,
Φp (a1 , . . . , an ) = (ap1 , . . . , apn ) for ∀a = (a1 , . . . , an ) ∈ Fq 24
n
stands for the Frobenius automorphism of degree p = charFq . Straightforwardly, X ∂ϕi p b1 bn p = I(Φ−1 p (b))ij LFq (x1 ) . . . LFq (xn ) ∂xj b∈Φp (S)
for ∀1 ≤ i, j ≤ n, whereas
∂ϕi (a) = I(a)ij ∂xj
at ∀a ∈ S ⊆ Fnq .
Therefore I(a) ∈ GL(n, Fq ) is the matrix of the differential n
n
(dϕ)a : Ta (Fq , Fq ) −→ Tϕ(a) (Fq , Fq ) n ∂ with respect to the basis ∂x∂ j , 1 ≤ j ≤ n of Ta (Fq , Fq ) and the basis ∂y i n
a
ϕ(a)
,
1 ≤ i ≤ n of Tϕ(a) (Fq , Fq ).
n
Finally, note that an arbitrary affine variety X/Fq ⊂ Fq admits a bijective morphism Φq : X → X, which is not biregular (or an isomorphism), as far as its inverse map Φ−1 q : X → X is not a morphism. The Frobenius automorphism Φq restricts to bijective maps Φq : X(Fqm ) → X(Fqm ) of the finite sets X(Fqm ) of the Fqm -rational points of X and, in particular, to the identity Φq = Id : X(Fq ) → X(Fq ) of the Fq -rational points. One can view Φq : T (X, Fqm ) −→ T (X, Fqm ) as a non-linear Hamming isometry of the Zariski tangent bundles T (X, Fqm ) for all m ∈ N. Note that Φq : Ta (X, Fqm ) → TΦq (a) (X, Fqm ) interchanges the fibres over a ∈ X(Fqm ) \ X(Fq ) and acts on the fibres Φq : Ta (X, Fqm ) → Ta (X, Fqm ) over a ∈ X(Fq ).
References [1] D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms - An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer, 1997. [2] W. Decker, G.-M. Greuel, G. Pfister, Primary decomposition : algorithms and comparisons, in Algorithmic Algebra an Number Theory (eds. G.-M. Greuel, B.H. Matzat and G. Hiss ), Springer, (1998), 187–220. [3] M.D. Fried, D. Haran and M. Jarden Effective c ounting of the points of definable sets over finite fields Israel Journal of Mathematics , 85 (1994), 103–133. [4] S. Lang and A. Weil Number of points of varieties in finite fields, Journal of Mathematics, 76 (1954), 819–827.
American
[5] W. Lizt, Die Anzahl der rationalen Punkte von Variet¨ aten u ¨ber einem endlichen K¨ orper,, Diplomarbeit, Heidelberg, 1975.
25
