Number of rational points of symmetric complete intersections over a finite field and applications
We study the set of common –rational zeros of systems of multivariate symmetric polynomials with coefficients in a finite field . We establish certain properties on these polynomials which imply that the corresponding set of zeros over the algebraic closure of is a complete intersection with “good” behavior at infinity, whose singular locus has a codimension at least two or three. These results are used to estimate the number of –rational points of the corresponding complete intersections. Finally, we illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.
Key words and phrases:Finite fields, symmetric polynomials, complete intersections, singular locus, factorization patterns, value sets
Several problems of coding theory, cryptography or combinatorics require the study of the set of rational points of varieties defined over a finite field on which the symmetric group of permutations of the coordinates acts. In coding theory, deep holes in the standard Reed–Solomon code over can be expressed by the set of zeros with coefficients in of certain symmetric polynomial (see, e.g., [CM07b] or [CMP12]). Further, the study of the set of –rational zeros of a certain class of symmetric polynomials is fundamental for the decoding algorithm for the standard Reed–Solomon code over of [Sid94]. In cryptography, the characterization of monomials defining an almost perfect nonlinear polynomial or a differentially uniform mapping can be reduced to estimate the number of –rational zeros of certain symmetric polynomials (see, e.g., [Rod09] or [AR10]). In [GM15], an optimal representation for the set of –rational points of the trace–zero variety of an elliptic curve defined over has been obtained by means of symmetric polynomials. Finally, it is also worth mentioning that applications in combinatorics over finite fields, as the determination of the average cardinality of the value set and the distribution of factorization patterns of families of univariate polynomials with coefficients in , has also been expressed in terms of the number of common –rational zeros of symmetric polynomials defined over (see [CMPP14] and [CMP15b]).
In [CMP12], [CMPP14] and [CMP15b] we have developed a methodology to deal with some of the problems mentioned above. This methodology relies on the study of the geometry of the set of common zeros of the symmetric polynomials under consideration over the algebraic closure of . By means of such a study we have been able to prove that in all the cases under consideration the set of common zeros in of the corresponding symmetric polynomials is a complete intersection, whose singular locus has a “controlled” dimension. This has allowed us to apply certain explicit estimates on the number of –rational zeros of projective complete intersections defined over (see, e.g., [GL02a], [CM07a], [CMP15a] or [MPP16]) to obtain a conclusion for the problem under consideration.
The analysis of [CMP12], [CMPP14] and [CMP15b] has several points in common, which may be put on a common basis that might be useful for other problems. For this reason, in this paper we present a unified and generalized framework where a study of the geometry of complete intersections defined by symmetric polynomials with coefficients in can be carried out along the lines of the papers above. More precisely, let be indeterminates over and let . Consider the weight defined by setting for and denote by the components of highest weight of . Let and the Jacobian matrices of and with respect to respectively. Suppose that
has full rank on every point of the affine variety ,
has full rank on every point of the affine variety .
Finally, if are new indeterminates over and are the first elementary symmetric polynomials of , we consider the polynomials defined as
We shall prove that the affine variety is a complete intersection whose geometry can be studied with similar arguments as those in the papers cited above. Further, we shall show how estimates on the number of –rational points of projective complete intersections can be applied in this unified framework to estimate the number of –rational points of . As a consequence, we obtain the following result.
Let assumptions and notations be as above. Denote and . If is the cardinality of the set of –rational points of , then the following estimate holds:
To illustrate the applications of Theorem 1.1, we shall consider two classical combinatorial problems over finite fields. The first one is concerned with the distribution of factorization patterns on a linear family of monic polynomials of given degree of . If the linear family under consideration consists of the monic polynomials of degree for which the first coefficients are fixed, then we have a function–field analogous to the classical conjecture on the number of primes in short intervals, which has been the subject of several articles (see, e.g., [Pol13], [BBR15] or [MP13, §3.5]). The study of general linear families of polynomial goes back at least to [Coh72]. Here, following the approach of [CMP15b], we obtain an explicit estimate on the number of elements on a linear family of monic polynomials of fixed degree of having a given factorization pattern, for which we rely on Theorem 1.1.
The second problem we consider is that of estimating the average cardinality of the value sets of families of polynomials of with certain consecutive coefficients prescribed. The study of the cardinality of value sets of univariate polynomials is the subject of the seminal paper [BS59]. In [Uch55] and [Coh73] the authors determine the average cardinality of the value set of all monic polynomials in of given degree, while [Uch55] and [Coh72] are concerned with the asymptotic behavior of the average cardinality of the value set of all polynomials of of given degree were certain consecutive coefficients are fixed. We shall follow the approach of [CMPP14], where the question is expressed in terms of the number of –rational points of certain complete intersection defined by symmetric polynomials.
The paper is organized as follows. In Section 2 we briefly recall the notions and notations of algebraic geometry we use. Section 3 is devoted to present our unified framework and to establish several results on the geometry of the affine complete intersections of interest. In Section 4 we study the behavior of these complete intersections “at infinity” and prove Theorem 1.1. Finally, in Sections 5 and 6 we apply Theorem 1.1 to determine the distribution of factorization patterns and the average cardinality of value sets of families of univariate polynomials.
2. Notions, notations and preliminary results
Let be any of the fields or . We denote by the –dimensional affine space and by the –dimensional projective space over . By a projective variety defined over (or a projective –variety for short) we mean a subset of common zeros of homogeneous polynomials . Correspondingly, an affine variety of defined over (or an affine –variety) is the set of common zeros in of polynomials . We shall frequently denote by or the affine or projective –variety consisting of the common zeros of the polynomials .
In what follows, unless otherwise stated, all results referring to varieties in general should be understood as valid for both projective and affine varieties. A –variety is –irreducible if it cannot be expressed as a finite union of proper –subvarieties of . Further, is absolutely irreducible if it is –irreducible as a –variety. Any –variety can be expressed as an irredundant union of irreducible (absolutely irreducible) –varieties, unique up to reordering, which are called the irreducible (absolutely irreducible) –components of .
For a –variety contained in or , we denote by its defining ideal, namely the set of polynomials of , or of , vanishing on . The coordinate ring of is defined as the quotient ring or . The dimension of is the length of the longest chain of nonempty irreducible –varieties contained in . We say that has pure dimension if all the irreducible –components of are of dimension .
The degree of an irreducible -variety is the maximum number of points lying in the intersection of with a linear space of codimension , for which is a finite set. More generally, following [Hei83] (see also [Ful84]), if is the decomposition of into irreducible –components, we define the degree of as
Let be a –variety and its defining ideal. Let be a point of . The dimension of at is the maximum of the dimensions of the irreducible –components of that contain . If , the tangent space to at is the kernel of the Jacobian matrix of the polynomials with respect to at . The point is regular if . Otherwise, the point is called singular. The set of singular points of is the singular locus of ; a variety is called nonsingular if its singular locus is empty. For a projective variety, the concepts of tangent space, regular and singular point can be defined by considering an affine neighborhood of the point under consideration.
Let and be irreducible affine –varieties of the same dimension and a regular map for which , where is the closure of with respect to the Zariski topology of . Such a map is called dominant. Then induces a ring extension by composition with . We say that the dominant map is a finite morphism if this extension is integral. We observe that the preimage of an irreducible closed subset under a dominant finite morphism is of pure dimension (see, e.g., [Dan94, §4.2, Proposition]).
Elements in or form a regular sequence if is nonzero and no is a zero divisor in the quotient ring or for . In that case, the (affine or projective) –variety is called a set–theoretic complete intersection. Furthermore, is called an (ideal–theoretic) complete intersection if its ideal over can be generated by polynomials. If is a complete intersection of dimension defined over , and is a system of homogeneous generators of , the degrees depend only on and not on the system of generators. Arranging the in such a way that , we call the multidegree of . According to the Bézout theorem (see, e.g., [Har92, Theorem 18.3]), in such a case we have .
In what follows we shall deal with a particular class of complete intersections, which we now define. A –variety is regular in codimension if its singular locus has codimension at least in , i.e., . A complete intersection which is regular in codimension 1 is called normal (actually, normality is a general notion that agrees on complete intersections with the one we define here). A fundamental result for projective normal complete intersections is the Hartshorne connectedness theorem (see, e.g., [Kun85, Theorem VI.4.2]), which we now state. If is a complete intersection defined over and is any –subvariety of codimension at least 2, then is connected in the Zariski topology of over . Applying the Hartshorne connectedness theorem with , one deduces the following result.
If is a normal complete intersection, then is absolutely irreducible.
2.1. Rational points
We denote by the –dimensional –vector space and by the set of lines of the –dimensional –vector space . For a projective variety or an affine variety , we denote by the set of –rational points of , namely in the projective case and in the affine case.
3. On the geometry of symmetric complete intersections
Let be positive integers with . Let be indeterminates over and let be the ring of polynomials in and coefficients in . We shall consider the weight on defined by setting for . Let and let be the affine –variety that they define. Let be the Jacobian matrix of with respect to . Assume that satisfy the following conditions:
form a regular sequence of ;
has full rank for every ;
The components of highest weight of satisfy () and ().
A polynomial is called weighted homogeneous (for the grading defined by the weight ) if all the monomials arising in this dense representation have the same weight. In this sense, it is clear that are weighted homogeneous.
In the next two remarks we show that polynomials as above, such that and satisfy , necessarily satisfy and . Nevertheless, as we shall rely repeatedly on the hypotheses , and , for the sake of readability we express the conditions that the polynomials must satisfy in this way.
If satisfy , then form a regular sequence of . Indeed, denote by the affine variety defined by and let be an arbitrary absolutely irreducible component of . Then . On the other hand, if is a regular point of , then the fact that satisfy implies that the tangent space of at has dimension at most . We conclude that . In other words, is of pure dimension . Finally, as are weighted homogeneous, the remark follows.
If and satisfy , then form a regular sequence of . Indeed, let be the homogenizations of with respect to the weight . We claim that the affine variety is of pure dimension .
To show the claim, let be an absolutely irreducible component of . It is clear that . Now let be a regular point of . Without loss of generality we may assume that either or . In the first case, the fact that the polynomials satisfy implies that . On the other hand, if , then , and since satisfy , we deduce that . In either case , which proves that . This finishes the proof of the claim.
Combining the claim with the fact that are weighted homogeneous, we conclude that form a regular sequence. In particular, is of pure dimension for . As the affine variety defined by the homogenization of each element of the ideal with respect to the grading defined by has dimension and is contained in the affine variety defined by for , we have for . This proves that form a regular sequence.
From () we conclude that the variety defined by is a set–theoretic complete intersection of dimension . Furthermore, by () it follows that the subvariety of defined by the set of common zeros of the maximal minors of the Jacobian matrix has codimension at least one. Then [Eis95, Theorem 18.15] proves that define a radical ideal, which implies that is a complete intersection.
Let be indeterminates over and the ring of polynomials in with coefficients in . Denote by the first elementary symmetric polynomials of . Let be the polynomials
Let for . In what follows we shall prove several facts concerning to the geometry of the affine –variety defined by .
For this purpose, consider the following surjective morphism of affine –varieties:
It is easy to see that is a finite morphism. In particular, the preimage of an irreducible affine variety of dimension is of pure dimension .
We now consider the polynomials as elements of , and denote . Observe that . Since form a regular sequence of , the variety has pure dimension for . This implies that the variety defined by has pure dimension for . Hence, the polynomials form a regular sequence of and we have the following result.
Let be the affine –variety defined by . Then is a set–theoretic complete intersection of dimension .
Next we study the singular locus of . To do this, we consider the following morphism of –varieties:
For and , we denote by and the tangent spaces to at and to at respectively. We also consider the differential map of at , namely
where is the following –matrix:
The main result of this section is an upper bound on the dimension of the singular locus of . To prove such a bound, we make some remarks concerning the Jacobian matrix of the elementary symmetric polynomials. It is well–known that the partial derivatives of the elementary symmetric polynomials satisfy the following identities (see, e.g., [LP02]) for :
As a consequence, if denotes the –Vandermonde matrix
then the Jacobian matrix of with respect to can be factored as follows:
Observe that is a square, lower–triangular matrix whose determinant is equal to . This implies that the determinant of is equal, up to a sign, to the determinant of , namely,
Denote by the Jacobian matrix of with respect to . The following result generalizes [CMPP14, Theorem 3.2].
The set of points for which does not have full rank, has dimension at most . In particular, the singular locus of has dimension at most .
By the chain rule, the partial derivatives of satisfy the following equality:
Fix an arbitrary point for which does not have full rank. Let be a nonzero vector in the left kernel of . Thus
where is the matrix defined in (3.2). Since by hypothesis () the Jacobian matrix has full rank, we deduce that is a nonzero vector with . Hence, all the maximal minors of must be zero.
Observe that is the –submatrix of which consists of the first rows of . Therefore, according to (3.3) we conclude that
where is the –submatrix of consisting of the first rows of . Since the last columns of are zero, we may rewrite this identity as follows:
where is the –submatrix of consisting of the first rows and the first columns of .
Since (3.5) holds for every as above, we conclude that has at least pairwise–distinct coordinates. In particular, the set of points for which is contained in a finite union of linear varieties of of dimension , and thus is an affine variety of dimension at most .
Now let be an arbitrary point of . By Lemma 3.4 we have that . Thus, the rank of is less than , for otherwise we would have , contradicting thus the fact that is a singular point of . This finishes the proof of the theorem. ∎
From the proof of Theorem 3.5 we conclude that the singular locus of is included in a simple variety of “low” dimension.
From Lemma 3.4 and Theorem 3.5 we obtain further consequences concerning the polynomials and the variety . According to Theorem 3.5, the set of points for which the matrix does not have full rank, has dimension at most . Since form a regular regular sequence and , [Eis95, Theorem 18.15] shows that define a radical ideal of , and thus is a complete intersection. Finally, the Bézout inequality (2.1) implies . In other words, we have the following statement.
The polynomials define a radical ideal and the variety is a complete intersection of degree at most .
4. The number of rational points of symmetric complete intersections
The results of the previous section on the geometry of the affine –variety , which is defined by the symmetric polynomials of (3.1), form the basis of our approach to estimate the number of –rational points of . As we shall rely on estimates for projective complete intersections defined over , we shall also need information on the behavior of “at infinity”.
4.1. The geometry of the projective closure
Consider the embedding of into the projective space which assigns to any the point . Then the closure of the image of under this embedding in the Zariski topology of is called the projective closure of . The points of lying in the hyperplane are called the points of at infinity.
It is well–known that is the –variety of defined by the homogenization of each polynomial belonging to the ideal (see, e.g., [Kun85, §I.5, Exercise 6]). Denote by the ideal generated by all the polynomials with . Since is radical it turns out that is also a radical ideal (see, e.g., [Kun85, §I.5, Exercise 6]). Furthermore, has pure dimension (see, e.g., [Kun85, Propositions I.5.17 and II.4.1]) and degree equal to (see, e.g., [CGH91, Proposition 1.11]).
Now we discuss the behavior of at infinity. Consider the decomposition of each polynomial into its homogeneous components, namely
where each is homogeneous of degree or zero, being nonzero for . Hence, the homogenization of each is the polynomial
It follows that for . Next we relate each with the component of highest weight of . Indeed, let be an arbitrary monomial arising in the dense representation of . Then its weight equals the degree of the corresponding monomial of . Hence, we easily deduce the following result.
Let be the homogeneous component of highest degree of and the component of highest weight of . Then for .
Let be the singular locus of at infinity, namely the set of singular points of lying in the hyperplane . From Lemma 4.1 we obtain critical information about .
The singular locus at infinity has dimension at most .
Let be an arbitrary point