The distribution of factorization patterns on linear families of polynomials over a finite field
We obtain estimates on the number of elements on a linear family of monic polynomials of of degree having factorization pattern . We show that , where is the proportion of elements of the symmetric group of elements with cycle pattern and is the codimension of . Furthermore, if the family under consideration is “sparse”, then . Our estimates hold for fields of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the –notation in terms of and with “good” behavior. Our approach reduces the question to estimate the number of –rational points of certain families of complete intersections defined over . Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of –rational points are established.
Key words and phrases:Finite fields, factorization patterns, symmetric polynomials, singular complete intersections, rational points
1991 Mathematics Subject Classification:12E05, 11T06, 12E20, 11G25, 14G05, 14G15, 14B05
Let be the finite field of elements, where is a prime number, and let denote its algebraic closure. Let be an indeterminate over and the set of polynomials in with coefficients in . Let be a positive integer and the set of all monic polynomials in of degree . Let be nonnegative integers such that
We denote by the set of elements with factorization pattern , namely the elements which have exactly monic irreducible factors over of degree (counted with multiplicity) for . We shall further use the notation for any subset .
In [Coh70] it was noted that the proportion of elements of in is roughly the proportion of permutations with cycle pattern in the th symmetric group . More precisely, it was shown that
where the constant underlying the –notation depends only on . A permutation of has cycle pattern if it has exactly cycles of length for . Observe that
In particular, is the number of permutations in with cycle pattern .
Furthermore, in [Coh72] a subset is called uniformly distributed if the proportion is roughly for every factorization pattern . The main result of this paper ([Coh72, Theorem 3]) provides a criterion for a linear family of polynomials of to be uniformly distributed in the sense above. As a particular case we have the classical case of polynomials with prescribed coefficients, where simpler conditions are obtained (see [Coh72, Theorem 1]; see also [Ste87]).
A difficulty with [Coh72, Theorem 3] is that the hypotheses for a linear family of to be uniformly distributed seem complicated and not easy to verify. In fact, in [GHP99] it is asserted that “more work need to be done to simplify Cohen’s conditions”. A second concern is that [Coh72, Theorem 3] imposes restrictions on the characteristic of which may inhibit its application to fields of small characteristic. Finally, we are also interested in finding explicit estimates, namely an explicit admissible expression for the constant underlying (1.1).
In this paper we consider the linear families in that we now describe. Let , be positive integers with , let be indeterminates over , and let be given linear forms of which are linearly independent and . Set and define as
Our main results assert that any such family is uniformly distributed. More precisely, we have the following result.
Let . If , and , then
On the other hand, if and , then
Here and are certain explicit discrete invariants associated to the linear variety under consideration. We have the worst–case upper bounds and .
Let and . We have the following result.
If , and , then
On the other hand, for and , we have
Theorem 1.1 strengthens (1.1) in several aspects. First of all, the hypotheses on the linear families in the statement of Theorem 1.1 are relatively wide and easy to verify. On the other hand, our results are valid either for or without any restriction on the characteristic of , while (1.1) requires that is large enough. A third aspect it is worth mentioning is that (1.4) shows that , while (1.1) only asserts that . Finally, both (1.3) and (1.4) provide explicit expressions for the constants underlying the –notation in (1.1) with a good behavior.
In order to prove Theorem 1.1, we express the number of polynomials in with factorization pattern in terms of the number of –rational solutions with pairwise–distinct coordinates of a system , where are certain polynomials in . A critical point for our approach is that, up to a linear change of coordinates, are symmetric polynomials, namely invariant under any permutation of . More precisely, we prove that each can be expressed as a polynomial in the first elementary symmetric polynomials of (Corollary 2.4). This allows us to establish a number of facts concerning the geometry of the set of solutions of such a polynomial system (see, e.g., Theorems 3.7, 3.11 and 5.1 and Corollary 5.2). Combining these results with estimates on the number of –rational points of singular complete intersections of [CMP12a], we obtain our main results (Theorems 4.2 and 5.4).
Our methodology differs significantly from that employed in [Coh70] and [Coh72], as we express in terms of the number of –rational points of certain singular complete intersections defined over . In [GHP99, Problem 2.2], the authors ask for estimates on the number of elements of , with a given factorization pattern, lying in nonlinear families of polynomials parameterized by an affine variety defined over . As a consequence of general results by [CvM92] and [FHJ94], it is known that , where is the dimension of the parameterizing affine variety under consideration. Nevertheless, very little is known on the asymptotic behavior of as a power of and of the size of the constant underlying the –notation. We think that our methods may be extended to deal with this more general case, at least for certain classes of parameterizing affine varieties.
2. Factorization patterns and roots
As before, let be a positive integer with and let be the set of monic polynomials of of degree . Let be the linear family defined in (1.2) and a factorization pattern. In this section we show that the number can be expressed in terms of the number of common –rational zeros of certain polynomials .
For this purpose, let be an arbitrary element of and let be a monic irreducible factor of of degree . Then is the minimal polynomial of a root of with . Denote by the Galois group of over . Then we may express in the following way:
Hence, each irreducible factor of is uniquely determined by a root of (and its orbit under the action of the Galois group of over ), and this root belongs to a field extension of of degree . Now, for a polynomial , there are roots of in , say (counted with multiplicity), which are associated with the irreducible factors of in of degree 1; it is also possible to choose roots of in (counted with multiplicity), say , which are associated with the irreducible factors of of degree 2, and so on. From now on we shall assume that a choice of roots of in is made in such a way that each monic irreducible factor of in is associated with one and only one of these roots.
Our aim is to express the factorization of into irreducible factors in in terms of the coordinates of the chosen roots of with respect to suitable bases of the corresponding extensions as –vector spaces. For this purpose, we express the root associated with each irreducible factor of of degree in a normal basis of the field extension .
Let be a normal element and let be the normal basis of generated by , namely
Observe that the Galois group is cyclic and the Frobenius map , is a generator of . Thus, the coordinates in the basis of all the elements in the orbit of a root of an irreducible factor of of degree are the cyclic permutations of the coordinates of in the basis .
The vector that gathers the coordinates of all the roots we have chosen to represent the irreducible factors of in the normal bases is an element of , which is denoted by . Set
for and . Observe that the vector of coordinates of a root is the sub-array of . With this notation, the irreducible factors of of degree are the polynomials
for . In particular,
Let be indeterminates over , set and consider the polynomial defined as
where the are defined as in (2.1). Our previous arguments show that an element has factorization pattern if and only if there exists with .
Next we discuss how many elements yield an arbitrary polynomial . For , we have that if and only if its orbit under the action of the Galois group has exactly elements. In particular, if is expressed by its coordinate vector in the normal basis , then the coordinate vectors of the elements of the orbit of form a cycle of length , because permutes cyclically the coordinates. As a consequence, there is a bijection between cycles of length in and elements with .
In this setting, the notion of an array of type will prove to be useful.
Let be defined as in (2.1). An element is said to be of type if and only if each sub-array is a cycle of length .
The next result relates with the set of elements of of type .
For any , the polynomial has factorization pattern if and only if is of type . Furthermore, for each square–free polynomial there are different with .
Let be the normal bases introduced before. Each is associated with a unique finite sequence of elements as follows: each with is the element of whose coordinate vector in the basis is the sub-array of .
Suppose that has factorization pattern for a given . Fix with and . Then is factored as in (2.2)–(2.3), where each is irreducible, and hence . We conclude that the sub-array defining is a cycle of length . This proves that is of type .
On the other hand, assume that we are given of type and fix with and . Then , because the sub-array is a cycle of length and thus the orbit of under the action of has elements. This implies that the factor of defined as in (2.2) is irreducible of degree . We deduce that has factorization pattern .
Furthermore, for of type , the polynomial is square–free if and only if all the roots with are pairwise–distinct, non–conjugated elements of . This implies that no cyclic permutation of a sub-array with agrees with another cyclic permutation of another sub-array . As cyclic permutations of any of these sub-arrays and permutations of these sub-arrays yield elements of associated with the same polynomial , we conclude that there are different elements with . ∎
2.1. in terms of the elementary symmetric polynomials
Consider the polynomial of (2.4) as an element of . We shall express the coefficients of by means of the vector of linear forms defined in the following way:
where is the matrix
According to (2.4), we may express the polynomial as
where are the elementary symmetric polynomials of . By the expression of in (2.4) we deduce that belongs to , which in particular implies that belongs to for . Combining these arguments with Lemma 2.2 we obtain the following result.
A polynomial has factorization pattern if and only if there exists of type such that
In particular, if is square–free, then there are elements for which (2.6) holds.
An easy consequence of this result is that we may express the condition that an element of has factorization pattern in terms of the elementary symmetric polynomials of .
A polynomial has factorization pattern if and only if there exists of type such that
In particular, if is square–free, then there are elements for which (2.7) holds.
3. The geometry of the set of zeros of
Let , and be positive integers with and . Given a factorization pattern , consider the family of monic polynomials of degree having factorization pattern , where is the linear family defined in (1.2). In Corollary 2.4 we associate to the following polynomials of :
The set of common –rational zeros of are relevant for our purposes.
Up to the linear change of coordinates defined by , we may express each as a linear polynomial in the first elementary symmetric polynomials of . More precisely, let be new indeterminates over . Then we have that
where are elements of degree whose homogeneous components of degree 1 are linearly independent in , namely the Jacobian matrix of with respect to has full rank .
In this section we obtain critical information on the geometry of the set of common zeros of the polynomials that will allow us to establish estimates on their number of common –rational zeros.
3.1. Notions of algebraic geometry
Since our approach relies on tools of algebraic geometry, we briefly collect the basic definitions and facts that we need in the sequel. We use standard notions and notations of algebraic geometry, which can be found in, e.g., [Kun85, Sha94].
We denote by the affine –dimensional space and by the projective –dimensional space over . Both spaces are endowed with their respective Zariski topologies, for which a closed set is the zero locus of polynomials of or of homogeneous polynomials of . For or , we say that a subset is an affine –definable variety (or simply affine –variety) if it is the set of common zeros in of polynomials . Correspondingly, a projective –variety is the set of common zeros in of a family of homogeneous polynomials . We shall frequently denote by the affine or projective –variety consisting of the common zeros of polynomials . The set is the set of –rational points of .
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 a -variety is the length of the longest chain of nonempty irreducible -varieties contained in . A –variety is called equidimensional if all the irreducible –components of are of the same 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 and be irreducible affine –varieties of the same dimension and let be a regular map for which holds, where denotes 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, namely if each element satisfies a monic equation with coefficients in . A basic fact is that a dominant finite morphism is necessarily closed. Another fact concerning dominant finite morphisms we shall use in the sequel is that the preimage of an irreducible closed subset is equidimensional of dimension (see, e.g., [Dan94, §4.2, Proposition]).
Let be a variety and let be the defining ideal of . 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 with respect to at . The point is regular if holds. 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.
Elements in or in form a regular sequence if is nonzero and each is not a zero divisor in the quotient ring or for . In such a case, the (affine or projective) variety they define is equidimensional of dimension , and is called a set–theoretic complete intersection. If, in addition, the ideal generated by is radical, then is an ideal–theoretic complete intersection. If is an ideal–theoretic complete intersection of dimension , 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 . The so–called Bézout theorem (see, e.g., [Har92, Theorem 18.3]) asserts that
In what follows we shall deal with a particular class of complete intersections, which we now define. A variety is regular in codimension if the singular locus of has codimension at least in , namely if . 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 defined here). A fundamental result for projective complete intersections is the Hartshorne connectedness theorem (see, e.g., [Kun85, Theorem VI.4.2]), which we now state. If is a set–theoretic complete intersection and is any subvariety of codimension at least 2, then is connected in the Zariski topology of . Applying the Hartshorne connectedness theorem with , one deduces the following result.
If is a normal set–theoretic complete intersection, then is absolutely irreducible.
3.2. The singular locus of the variety
With the notations and assumptions of the beginning of Section 3, let be the affine variety defined by the polynomials of (3.1). The main result of this section asserts that is regular in codimension one. From this result we will be able to conclude that is a normal ideal–theoretic complete intersection.
In the sequel we shall frequently express the points of in the coordinate system , where are the linear forms of (2.5). Let be new indeterminates over , set and let be the linear polynomials for which holds for , where are the first elementary symmetric polynomials of . Recall that, by hypothesis, the Jacobian matrix of with respect to has full rank .
We now consider as elements of . Since the Jacobian matrix has full rank , the linear affine variety that define has dimension . Consider the following surjective mapping:
It is easy to see that is a dominant finite morphism (see, e.g., [Sha94, §5.3, Example 1]). In particular, the preimage of an irreducible affine variety of dimension is equidimensional and of dimension .
Observe that the affine linear variety is equidimensional of dimension . This implies that the affine variety is equidimensional of dimension . We conclude that form a regular sequence of and deduce the following result.
Let be the affine variety defined by . Then is a set–theoretic complete intersection of dimension .
Next we analyze the dimension of the singular locus of . Assume without loss of generality that is lower triangular in row–echelon form. Let be the indices corresponding to the pivots. Let and . Then the Jacobian matrix
is invertible. Let be new indeterminates over and define and . Set , and . Observe that the Jacobian matrix
is also invertible. Consider the following surjective morphism of affine varieties: