Shifted varieties and discrete neighborhoods around varieties

12/17/2018 ∙ by Joachim von zur Gathen, et al. ∙ Universidad Nacional de General Sarmiento University of Bonn 0

For an affine variety X defined over a finite prime field F_p and some integer h, we consider the discrete h-neighborhood of the set of F_p-rational points, consisting of those points over F_p whose distance from X is not more than h, for a natural notion of "distance". There is a natural upper bound on its size. We address the question whether the neighborhood's size is close to its upper bound. The central notion for understanding this question turns out to be the shift of a variety, which is the translation by a nonzero constant vector of the coordinates. If no absolutely irreducible component with maximal dimension of X is a shift of another component, then the answer to the question is "yes". For the opposite case, we exhibit examples where the answer is "no". When X is absolutely irreducible, the condition on shifts turns out to be necessary and sufficient. Computationally, testing the condition is coNP-complete under randomized reductions, already for simple cases.

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1. Introduction

We take an odd prime

, the field with elements considered as a subset of the integers with arithmetic modulo , positive integers and , the -norm for and the ball of radius and with elements. Furthermore, is an algebraic closure of .

Given a system of polynomials in , we consider its affine variety , its rational points that lie in , and the “discrete neighborhood” around of radius . Then

(1.1)

Question (Q) asks for upper bounds on the (non-negative) difference

(1.2)

A small such bound implies a lower bound on . Under mild assumptions, we show that it is close to the upper bound in some cases, and we also exhibit other cases where the gap is substantial. Testing whether the assumption holds for a given variety turns out to be computationally infeasible.

2. Preliminaries

Let be a power of a prime and a finite field with elements. We denote by the algebraic closure of and by the affine -dimensional space over . A nonempty subset is an affine subvariety of (a variety for short) if it is the set of common zeros in of some set of polynomials in . Further, is an -variety (or -definable) if it can be defined by polynomials in . We will use the notations and to denote the -variety defined by .

An -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 an –variety. Any -variety can be expressed as a non-redundant union of irreducible -varieties, unique up to reordering, which are called the irreducible -components of . Some of them may be absolutely irreducible.

For an -variety , its defining ideal is the set of polynomials in vanishing on . The dimension of an -variety is the length of a longest chain of nonempty irreducible -varieties contained in .

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 finite. More generally, following Heintz83 (see also Fulton84), if is the decomposition of into irreducible -components, then the degree of is

The following Bézout inequality holds (see Heintz83, Fulton84, Vogel84): if and are -varieties, then

(2.1)

In the following, we usually state explicit inequalities, but the spirit is that the field size is (much) larger than the geometric quantities like , , and , so that our bounds should be taken as asymptotics in . Thus an upper bound on is “small” if it is of smaller order in than the arguments of .

We denote by the set of -rational points of an -variety , namely, . For of dimension and degree , we let be the decomposition of into irreducible -components, and suppose that are absolutely irreducible of dimension and that are not. Then provide the main contribution to and we call them the essential components. We write

Then the following bounds on hold.

Fact 2.1.
  1. If and , then

  2. If is an irreducible -variety and not absolutely irreducible, then

Proofs of (ii) and (iii) are in (CaMa06, Theorem 5.7 and Lemma 2.3).

3. Neighborhoods around varieties

Given a polynomial sequence in and the affine -variety , question (Q) is concerned with the size of the “standard neighborhood”

As is an optimal upper bound, we concentrate on lower bounds, or, equivalently, on upper bounds on the difference from (1.2).

3.1. Generalized neighborhoods around varieties

Most of this paper deals with the following more general problem: given an -variety , and a nonempty set , find lower bounds on

Since

(3.1)

it is sufficient to show upper bounds on the latter sum.

For with and with , we have . If , then with .

We fix the following notation for an irreducible decomposition of an -variety of dimension :

(3.2)

with . Recall that are the essential components. According to Fact 2.1, the cardinality of the set for an essential component is of order , while that of the other components is at most of order . The following notions of shifts are central to our considerations.

Definition 3.1.
  1. For and , is shifted by .

  2. For and , consists of the polynomials in shifted by , and is shifted by .

  3. with decomposition (3.2) is essentially -shift-free if none of its components equals for any with . (The case is included, and if , then is essentially -shift-free.)

  4. For , is essentially -shift-free if it is essentially -shift-free for all .

Thus is isomorphic to , and if , then . When is absolutely irreducible, so that in (3.2), we leave out the word “essentially”.

Lemma 3.2.

We have

Proof.

For with , with , and , we have and . Since any can be expressed in at most ways as with , the lemma follows. ∎

The following result first establishes an upper bound on the difference from (3.1) in terms of intersections of shifted varieties. Then we state, under a shift-freeness assumption, a bound which is small compared to the two arguments of .

Theorem 3.3.

Let be an -variety of dimension , degree , and decomposition (3.2).

  1. If furthermore is essentially -shift-free, then

Proof.

Inequality (3.1) and Lemma 3.2 imply that

Given

, we estimate

For and , we have . Similarly, holds for and all . Thus all these intersections are contained in . Together with this yields the bound claimed in (i).

For (ii), we first have for by Fact 2.1. It remains to consider . By hypothesis and are distinct absolutely irreducible varieties and their intersection has dimension at most and degree at most by the Bézout inequality (2.1). By Fact 2.1 (i) we have for any that

For we have the following less precise result, which follows from Fact 2.1 (i) and (iii), and (1.1) (for general ).

Corollary 3.4.

With hypotheses and notations as in Theorem 3.3, assume further that . Then

When is not shift-free, then may be large, as in the following example.

Example 3.5.

Let , , , , so that and is the union of

parallel hyperplanes

for with distance 1 between “neighbors”, and . is invariant under many shifts. Namely, for any , and for , for any . We have , , and

Since , there is no “small” upper bound on , namely of order less than in . In particular, if is a single hyperplane, then . Its difference with is of the same order of magnitude in as its two arguments, that is, not “small”.

Example 3.6.

Generalizing Example 3.5, we consider a variety whose defining polynomials are independent of the variables , for some with . We may also consider them as elements of , they define a variety , and . We consider the embedding

(3.3)

take some , and let . Then .

If we write and assume that for some with , then

(3.4)

We will modify this reasoning in Theorem 4.3 to show that under a certain condition, no “small” upper bound on exists.

Example 3.7.

For and , we consider the “determinantal” -variety of matrices in of rank at most . It is well-known that is absolutely irreducible with

see, e.g., (BrVe88, Proposition 1.1) for the first assertion and (Harris92, Example 19.10) for the second one. A simple calculation reveals that those factors decrease monotonically with growing , so that the term for dominates and .

In view of Theorem 3.3, we check that is not shift-invariant. Let and consider

. As the zero matrix

belongs to , if is in , then . Let

be an invertible matrix such that the last

rows of are equal to zero. Let be a matrix whose first rows and last rows are zero, and the remaining rows, together with the first rows of , are linearly independent. Such an exists, since . Then

It follows that and , so that , and thus is -shift-free. Applying Theorem 3.3 we obtain for

(3.5)

As a further example, we consider the variety of decomposable univariate polynomials. For a univariate polynomial in a polynomial ring over a ring and , its th Hasse derivative is

(3.6)

Since in the usual ranges for binomial coefficients, we have .

Example 3.8.

A univariate polynomial of degree over a field is decomposable if there exist of degrees , respectively, with . Then , and denoting their coefficients by and , respectively, we also have for the monic (leading coefficient 1) and original (constant coefficient 0, graph containing the origin) polynomial :

We may thus assume that is monic original. Then . We might further normalize into the monic original , so that in all three polynomials are monic original, with the appropriate , but do not use this here.

All such polynomials , , and are parametrized by their coefficients in , , and , respectively. The Zariski closure of the image of the composition map with is the set of decomposable polynomials. This is an absolutely irreducible closed affine subvariety of , of dimension and with degree ; see gatmat18.

In the remainder of this example, we assume that is a finite field of characteristic greater than and that is sufficiently large (compared to ).

We want to show that is -shift-free. It is sufficient to exhibit for every nonzero some so that . Addition here is the standard coefficient-wise addition of polynomials.

By the chain rule, the Hasse derivative

of any with , and has a factor of degree . We consider the set of polynomials of of degree at most . We claim that the set of having a factor in of degree is Zariski closed. Indeed, fix a factorization pattern for polynomials of degree , where are such that . For each , let be the product of all irreducible polynomials of of degree . Then has a factor in with factorization pattern if and only if has degree at least for . Further, the latter is equivalent to the vanishing of the first subresultants of and for . This shows that the set of elements of having a factor in with factorization pattern is Zariski closed. Considering all possible factorization patterns for polynomials of degree , the claim follows.

Since the set of all as above is Zariski dense in and each has a factor in of degree , we conclude that the derivatives of all satisfy this closed condition.

We consider a nonzero , supposing first that and . Any with belongs to and it suffices to prove that there exists such an with . When vary over , the set of polynomials

constitutes a linear family with prescribed coefficients in the sense of, e.g., (MuPa13, §3.5). Arguing by contradiction, assume that for any as above. Denote by the set of exponents corresponding to the monomials having a prescribed value. Then , and (Cohen72, Theorem 1) shows that for sufficiently large , there exists a polynomial so that is irreducible in . According to our previous remarks, this contradicts our assumption .

For the remaining case, where , we add a fixed term with and let the remaining terms vary, while if , we make a similar argument considering the Taylor expansion of in powers of and the set of elements with , for a suitable .

Thus is -shift-free and the estimate of Theorem 3.3 (ii) applies.

4. Shift-invariant varieties

In order to apply Theorem 3.3 (ii) to an absolutely irreducible -variety , the critical point is to check whether is -shift-free. We say for some that is -shift-invariant if . is shift-invariant if it is -shift-invariant for some nonzero . is a cylinder in the direction if for any and , . We have the following characterization of shift-invariance.

Proposition 4.1.

Let , a power of , let be an -variety of degree and . Then is -shift-invariant if and only if is a cylinder in the direction of .

Proof.

Suppose that is invariant under a shift . Let be an arbitrary point of and consider the line . Since is invariant under the shift , it is also invariant under . Thus

If , then by the Bézout inequality (2.1) we would have

which contradicts the previous inequality. It follows that

so that . The fact that the variety of dimension 1 is contained in the absolutely irreducible variety of dimension 1 shows that , that is, . Since this holds for any , we conclude that is a cylinder in the direction of .

The converse assertion is clear. ∎

We can reformulate the condition of shift-invariance as follows.

Corollary 4.2.

With hypotheses as in Proposition 4.1, is shift-invariant if and only if there exists an invertible map of linear forms in such that for some -variety . If this is the case and is the ideal of , then .

Proof.

We assume that with some nonzero . For ease of presentation, we apply a coordinate permutation so that , and now assume . We define the vector of linear forms

(4.1)

and also denote the induced mapping as

The linear forms in are linearly independent. Let be followed by the projection to the first coordinates, and . Thus . We claim that .

The inclusion “” is clear. So let and , and for some . As in the proof of Proposition 4.1, the line is contained in . Then

which shows the claim. The invertible map in the proposition is .

Finally, the identity is a standard fact on ideals of varieties. ∎

In the next result, a subset is called closed under shifts to zero if for any , replacing any coordinate of by zero yields an element of . We remark that the standard neighborhood is closed under shifts to zero. Finally, we recall from (3.1).

Theorem 4.3.

Let be a prime, , closed under shifts to zero, and be an absolutely irreducible -variety of dimension and degree which is not -shift-free. Furthermore, let

(4.2)

and assume that . Then

(4.3)
Proof.

The mapping given by for an -variety constitutes an action of the additive group on the set of -varieties, since . We let be a basis of the subgroup generated by the with . This subgroup is an -vector space of some dimension and we write . We take the invertible linear map from (4.1) with , ignoring for ease of presentation the possibly required coordinate permutation. Thus and , for some subvariety of . Also, is