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Symmetric Functions Over Finite Fields

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Symmetric Functions Over Finite Fields Symmetric functions over finite fields Mihai Prunescu ∗ Abstract The number of linear independent algebraic relations among elementary symmetric poly- nomial functions over finite fields is computed. An algorithm able to find all such relations is described. The algorithm consists essentially of Gauss' upper triangular form algorithm. It is proved that the basis of the ideal of algebraic relations found by the algorithm consists of polynomials having coefficients in the prime field Fp. A.M.S.-Classification: 14-04, 15A03. 1 Introduction The problem of interpolation for symmetric functions over fields is an important question in Algebra, and a lot of aspects of this problem have been treated by many authors; see for example the monograph [2] for the big panorama of symmetric polynomials, or [3] for more special results concerning the interpolation. In the case of finite fields the problem of interpolation for symmetric functions is in the same time easier than but different from the general problem. It is easier because it can always be reduced to systems of linear equations. Indeed, there are only finitely many monomials leading to different polynomial functions, and only finitely many tuples to completely define a function. The reason making the problem different from the general one is not really deeper. Let Si be the elementary symmetric polynomials in variables Xi (i = 1; : : : ; n). If the exponents are bounded by q, one has exactly so many polynomials in Si as polynomials in Xi, but strictly less symmetric functions n from Fq ! Fq than general functions. It follows that a lot of polynomials in Si must express the constant 0 function. We call this set of polynomials the ideal of algebraic relations between elementary symmetric functions over the finite field Fq and denote this I(q; n). Every instance of the interpolation problem for a symmetric function has as set of solutions a class f0(S~) + I(q; n). In this paper we study the ideal of algebraic relations I(q; n) and we compute its dimension as a vector space over the finite field. The paper is organized as follows: In the Section 2 definitions and notations are rigorously given. The ideal I(q; n) is defined as the kernel of a morphism Φ. In the n Section 3 the number of symmetric functions Fq ! Fq is counted. Furtherly it is proved that the morphism Φ is surjective and the dimension of I(q; n) as a vector space over Fq is computed. As a byproduct of this Section, I(q; n) is also the kernel of a substitution morphism between two rings of polynomials. In the Section 3 we describe an algorithm for the automatic deduction of all such algebraic relations. The algorithm consists essentially of Gauss' upper triangular form algorithm to parametrically solve systems of linear equations. We prove that the algorithm always finds a basis of the vector space of algebraic relations, always consisting of polynomials with coefficients in the prime field Fp. We also display in the Section 4 some examples of algebraic relations produced by the algorithm. The Section 5 contains an example of concrete interpolation problem. This problem was the original motivation of the author to do these computations. ∗Brain Products, Freiburg, Germany, and Institute of Mathematics of the Romanian Academy, Bucharest, Romania. [email protected]. 1 2 Definitions and notations Consider a finite field Fq of characteristic p. The elements of Fq are identified with the set f0; 1; : : : ; q − 1g in an arbitrary way. For the rest of the paper we fix a natural number n ≥ 2 and two sets of variables: S1, ... , Sn and X1, ... , Xn. The variables Si will be in some contexts algebraically independent variables. In other contexts Si shall denote the projection of the variable Si in different homomorphic images of the polynomial ring Fq[S1;:::;Sn] or shall mean an elementary symmetric polynomial in variables X1;:::;Xn. α1 α2 αn Definition: The set Mon(q; n) is the set of all monomials S1 S2 :::Sn with 0 ≤ αi < q. There are qn many such monomials. Definition: Let FqfS1;:::;Sng be the vector space over Fq freely generated by the set Mon(q; n). n FqfS1;:::;Sng has dimension q over Fq. It has also a canonical structure of finite ring induced by the epimorphism: s : Fq[S1;:::;Sn] −! FqfS1;:::;Sng q q with Ker(s) = (S1 − S1;:::;Sn − Sn) as ideal in Fq[S1;:::;Sn]. This construction is justified by q the fact that Fq j= 8x x = x. Definition: Sym(m) denotes the symmetric group of all permutations of m objects f1; : : : mg. Definition: For every function f : f0; : : : ; q − 1gn ! f0; : : : ; q − 1g and permutation σ 2 Sym(n) σ we define f (x1; : : : ; xn) = f(σ(~x)) where σ(~x) = (xσ(1); : : : ; xσ(n)). The function f is called symmetric if for all σ 2 Sym(n), f = f σ. Definition: Remember here that Fq and the set f0; : : : ; q − 1g have been identified. Let F(q; n) n denote the set of all functions f : Fq ! Fq and S(q; n) ⊂ F(q; n) the subset of all symmetric functions. Both sets equiped with the point-wise operations are finite rings and finite vector spaces over Fq. σ ~ ~ Definition: For every F 2 Fq[X1;:::;Xn] and σ 2 Sym(n) we define F (X) = F (σ(X)), where ~ σ σ(X) = (Xσ(1);:::;Xσ(n)). F is called symmetric if for all σ 2 Sym(n), F = F . n Definition: For 0 ≤ k ≤ n denote by Pk the set of subsets of f1; : : : ; ng containing exactly k elements. Recall that the elementary symmetric polynomials Sk(X1;:::;Xn) are defined as: X Y Sk(X1;:::;Xn) = Xi: n J2Pk i2J Definition: Consider the function Φ: FqfS1;:::;Sng −! S(q; n) defined such that n ~ 8~a 2 Fq Φ(f(S))(~a) = f(S1(~a);:::;Sn(~a)); ~ where f(S) 2 FqfS1;:::;Sng. In the right hand side the variables Si are interpreted as elementary symmetric polynomials which are evaluated in the tuple ~a. Φ is a well defined homomorphism of finite rings and of finite vector spaces over Fq. Definition: The ideal I(q; n) = Ker(Φ) ⊂ FqfS1;:::;Sng is called ideal of algebraic relations between elementary symmetric functions over Fq. I(q; n) is also a sub-space of FqfS1;:::;Sng. The goal of this paper is to find out the dimension (cardinality) of I(q; n) and to describe a concrete method to find its the elements. 3 The number of algebraic relations The fixed identification of Fq and f0; : : : ; q − 1g is crucial for this section. 2 Definition: Let WM(q; n) be the set of all (weakly) monotone increasing tuples (a1; : : : ; an) with all ai 2 f0; : : : ; q − 1g. Denote by wm(q; n) the cardinality of the set WM(q; n). Lemma 3.1 n + q − 1 dim S(q; n) = wm(q; n) = : Fq q − 1 Proof: For the first equality: in order to define an f 2 S(q; n), it is enough to define its values for every w 2 WM(q; n). For the second equality: the number of partitions with at most k parts and the largest part ≤ j k+j is j , as proven for example in [1]. Now take k = n and j = q − 1. 2 n Definition: Consider the following matrix M(q; n) 2 Mat(wm(q; n)×q ; Fq). The rows of M(q; n) are indexed using the tuples ~ι 2 WM(q; n), the columns are indexed using the monomials m 2 Mon(q; n), and if M(q; n) = (a(~ι,m) j ~ι 2 WM(q; n); m 2 Mon(q; n)), a(~ι,m) = [Φ(m)](~ι): Theorem 3.2 The rank of the matrix M(q; n) is maximal: n + q − 1 rank M(q; n) = wm(q; n) = : q − 1 The dimension of the ideal I(q; n) of algebraic relations as a vector space over Fq is: n + q − 1 dim I(q; n) = qn − wm(q; n) = qn − : Fq q − 1 Proof: Linear algebra using Lemma 3.1 and the following Lemma 3.3. 2 Lemma 3.3 The morphism Φ: FqfS1;:::;Sng ! S(q; n) with Ker Φ = I(q; n) is surjective. Proof: The proof consists of two steps. In the first step we repeat the interpolation over finite fields and check that by interpolating symmetric functions the method produces symmetric polynomials. In the second step, we repeat the argument that a symmetric polynomial can be written as a polynomial in elementary symmetric polynomials and convince us that the existence part of the proof behaves well with the degrees. n Step 1: Let f : Fq ! Fq be some function, for the moment not necessarily symmetric. For a 2 Fq define the polynomial ha 2 Fq[X]: Y X − λ h (X) = : a a − λ λ2Fq nfag n Observe that ha(a) = 1 and ha(Fq n fag) = 0. For a tuple ~a 2 Fq define h~a 2 Fq[X1;:::;Xn]: ~ h~a(X) = ha1 (X1) : : : han (Xn): A polynomial interpolating f is: ~ X ~ H(X) = h~a(X)f(~a): n ~a2Fq σ σ We observe that for all σ 2 Sym(n), (h~a) = hσ−1(~a). If the function f is symmetric, then f = f and it follows: σ ~ X σ ~ X ~ −1 ~ H (X) = h~a (X)f(~a) = hσ−1(~a)(X)f(σ (~a)) = H(X): n −1 n ~a2Fq σ (~a)2Fq 3 We proved that the interpolation algorithm applied to a symmetric function leads to a symmetric polynomial.
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