aa r X i v : . [ m a t h . N T ] O c t POLYNOMIAL VALUES IN AFFINE SUBSPACES OFFINITE FIELDS
ALINA OSTAFE
Abstract.
In this paper we introduce a new approach and obtainnew results for the problem of studying polynomial images of affinesubspaces of finite fields. We improve and generalise several pre-vious known results, and also extend the range of such results topolynomials of degrees higher than the characteristic of the field.Such results have a wide scope of applications similar to those asso-ciated with their counterparts studying consecutive intervals overprime fields instead of affine subspaces. Here we give only two im-mediate consequences: to a bound on the size of the intersection oforbits of polynomial dynamical systems with affine subspaces andto the Waring problem in affine subspaces. These results are basedon estimates for a certain new type of exponential sums. Introduction
Motivation.
Given a polynomial f over a field F and two “interesting”finite sets A , B ⊆ F it is natural to ask about the size of the intersection f ( A ) ∩ B = { f ( a ) : a ∈ A and f ( a ) ∈ B} and in particular improve the trivial bound min { A , B} on the sizeof this intersection. In particular, for the case of prime finite fields F p of p elements with A , B chosen as intervals of consecutive elements (ina natural ordering of elements of F p ) a series of such results have beenobtained in [7, 8, 9] where also a broad variety of application has beengiven. For example, one of motivating applications for these resultscomes from the study of the number of points in polynomial orbitsthat fall in a given interval, see [5, 6, 9, 18].Here we mostly concentrate on the case of finite fields that are highdegree extensions of prime fields. Furthermore, our sets A , B are affinesubspaces which are natural analogues of intervals in these settings. Date : June 15, 2018.2010
Mathematics Subject Classification.
Key words and phrases. finite fields, exponential sums, polynomial dynamics,Waring problem.
More precisely, for a prime power q and an integer r > K = F q and L = F q r the finite fields of q and q r elements, respectively,and consider affine subspaces of L over K . We are especially interestedin the case when the dimension s of these spaces is small compared to r and thus standard approaches via algebraic geometry methods (suchas the Weil bound) do not apply.We note that a similar point of view has recently been accepted byCilleruelo and Shparlinski [10] and by Roche-Newton and Shparlin-ski [22] who obtained several results in this direction via the methodsof additive combinatorics. In fact, the results and method of [10] applyonly to a very special class of affine spaces, while [22] addresses the caseof arbitrary affine spaces. Here, using a different approach, we improvesome of the results of [22] and also obtain a series of other results. Inparticular, we obtain some nontrivial results for a class of polynomi-als of degree d ≥ p , where p is the characteristic of L , while for theinductive method of [22] the condition d < p seems to be unavoidable.More precisely, our approach appeals to the recent bounds of Bour-gain and Glibichuk [4] of multilinear exponential sums in arbitraryfinite fields which we couple with the classical van der Corput differ-encing. We use this combination to estimate exponential sums withpolynomials of degree d along affine spaces.We remark that under some natural conditions the dimension s ofthese spaces can be as low as r/d by the order of magnitude. Thiscorresponds exactly to the lowest possible length of intervals over F p for which one can estimate nontrivially the corresponding exponentialsums via Vinogradov’s method, see the recent striking results of Woo-ley [26, 27, 28].As in the previous works in this direction, we also give some appli-cations of our results.Namely, we study the intersection of orbits of polynomial dynamicalsystems and affine spaces and improve and complement some resultsof Roche-Newton and Shparlinski [22]; both are analogues of thoseof [5, 6, 9, 18]. We also recall that this question has been introducedby Silverman and Viray [23] in characteristic zero and then studiedusing a very different technique.Finally, we also consider the Waring problem in subspaces.We now outline in more detail our main results, that are given inTheorems 10, 20 and 22 below. Exponential sums over affine subspaces and polynomial valuesin affine subspaces.
Our first motivation is to estimate the numberof elements u in an affine subspace A of L , that is, a translate of a OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 3 linear subspace of L , such that f ( u ) falls also in an affine subspace B of L . We denote this number by I f ( A , B ), that is, for a nonlinearpolynomial f ∈ L [ X ],(1) I f ( A , B ) = { u ∈ A | f ( u ) ∈ B} . The basic tool in obtaining estimates for I f ( A , B ) is using a recentestimate of Bourgain and Glibichuk [4, Theorem 4] on multilinear ex-ponential sums over subsets of L , see Lemma 8 below. To arrive tousing this result, we apply the classical van der Corput differencingmethod for our exponential sum to reduce the degree of the polyno-mial f , see also [3, Theorem C]. However, this method was applied sofar only with polynomials of degree less than p .Let ψ be an additive character of L and χ : L → C a functionsatisfying χ ( x + y ) = χ ( x ) χ ( y ), x, y ∈ L . The first main result of thispaper is obtaining, under certain conditions, estimates for exponentialsums over affine subspaces A of L of the type X x ∈A χ ( x ) ψ ( f ( x )) . What is new about this result is that it applies to several classes ofpolynomials of degree larger than p or a multiple of p , see Theorem 10,in contrast to previous results that apply only to polynomial of degreeless than p .To estimate I f ( A , B ), we first use the classical Weil bound to esti-mate exponential sums, but the bound we obtain is nontrivial only for s > r (1 / ε ), for some ε >
0, and it also applies only for polyno-mials of degree less than p . However, applying Theorem 10 we obtainnontrivial estimates for any s ≥ εr , and moreover for more generalpolynomials of degree larger or equal to p , see Theorem 16.The bound of Theorem 16 improves the very recent estimate (17)obtained in [22] for s < . (cid:0) (cid:1) d rε . Moreover, Theorem 16 generalisesthe result of [22] as this holds only for polynomials of degree d < p .We also conclude from Theorem 16 that, under certain conditions, f ( A ) is not included in any proper affine subspace B of L , see Corol-lary 17. Polynomial orbits in affine subspaces.
Given a polynomial f ∈ L [ X ] and an element u ∈ L , we define the orbit(2) Orb f ( u ) = { f ( n ) ( u ) : n = 0 , , . . . } , where f ( n ) is the n th iterate of f , that is, f (0) = X, f ( n ) = f ( f ( n − ) , n ≥ . A. OSTAFE
As the orbit (2) is a subset of L , and thus a finite set, we denote by T f,u = f ( u ) to be the size of the orbit.Here we study the frequency of orbit elements that fall in an affinesubspace of L considered as a linear vector space over K . This questionis motivated by a recent work of Silverman and Viray [23] (in charac-teristic zero and using a very different technique). Recently, severalresults have been obtained in [22] using additive combinatorics. Herewe improve on several results of [22] and we also extend the class ofpolynomials to which these results apply, see Corollary 18.We also note that the argument of the proof of [22, Theorem 6] cangive information about the frequency of (not necessarily consecutive)iterates falling in a subspace. We present such a result in Theorem 20,as well as apply it to obtain information about intersection of orbits oflinearised polynomials in Corollary 21. Exponential sums over consecutive integers and the Waringproblem.
For a positive integer n ≤ p r −
1, we consider the p -adicrepresentation n = n + n p + . . . + n s − p s − for some s ≤ r . Let 1 ≤ N ≤ p r − f ∈ L [ X ] be a polynomialof degree d . Furthermore, let ψ be an additive character of L and let χ : N → C a p -multiplicative function, see Section 6 for a definition.Another main result of this paper is to estimate, using Theorem 10,under certain conditions, the twisted exponential sum S ( N ) = X n ≤ N χ ( n ) ψ ( f ( ξ n )) , where ω , . . . , ω r − is a basis of L over F p and ξ n = s − X i =0 n i ω i , which we hope to be of independent interest. We present such a resultin Theorem 22 using the class of p -multiplicative functions χ ( n ) = exp πi s − X j =0 α j n j ! , where α j , j = 0 , , . . . , is a fixed infinite sequence of real numbers.Let f ∈ L [ X ] be a polynomial of degree d . As another direct conse-quence of Theorems 10 we prove the existence of a positive integer k such that for any y ∈ L , the equation f ( ξ n ) + . . . + f ( ξ n k ) = y OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 5 is solvable in positive integers n , . . . , n k ≤ N . We do this first for thecase N = q s − q s − ≤ N < q s . Recently, quite substantial progress has beenachieved in the classical Waring problem in finite fields, see [11, 12, 13,25].We conclude the paper with some remarks and possible extensionsof our results, as well as some connections to constructing affine dis-persers. 2. Consecutive differences of polynomials
For our main results we need a few auxiliary results regarding consec-utive differences of polynomials. For a polynomial f ∈ L [ X ] of degree1 ≤ d < p with leading coefficient a d , we define∆ X ,X ( f ) = f ( X + X ) − f ( X ) = X f ( X , X ) , for some polynomial f ( X , X ) ∈ L [ X , X ] of degreedeg X f ( X , X ) = d − da d . Inductively, we define∆ X ,...,X k ( f )= f ( X , . . . , X k − , X k − + X k ) − f ( X , . . . , X k − , X k )= X k − f ( X , . . . , X k ) , (3)for some polynomial f ( X , . . . , X k ) ∈ L [ X , . . . , X k ] of degreedeg X k f = d − ( k − X k , d ( d − . . . ( d − ( k − a d .We also have the following relation(4)∆ X ,...,X k ( f ) = 1 X k − (cid:0) ∆ X ,...,X k − ,X k − + X k ( f ) − ∆ X ,...,X k − ,X k ( f ) (cid:1) . We now give more details in the following straightforward statementwhich is well-known but is not readily available in the literature.
Lemma 1.
Let f ∈ L [ X ] be a polynomial of degree d < p and leadingcoefficient a d ∈ L ∗ . Then ∆ X ,...,X k ( f ) = X k − f ( X , . . . , X k ) where f ( X , . . . , X k )= d ( d − . . . ( d − k + 2) a d X d − k +1 k + e f ( X , . . . , X k ) , (5) A. OSTAFE for some polynomial e f ( X , . . . , X k ) ∈ L [ X , . . . , X k ] of degrees deg X i e f ≤ d − k + 1 , i = 1 , . . . , k − , deg X k e f ≤ d − k. Proof.
The result follows by induction over k . Easy computations provethe statement for k = 2. We assume it is true for k − k . Using the induction hypothesis, we have∆ X ,...,X k ( f )= f ( X , . . . , X k − , X k − + X k ) − f ( X , . . . , X k − , X k )= d ( d − . . . ( d − k + 3) a d ( X k − + X k ) d − k +2 + e f ( X , . . . , X k − , X k − + X k ) − d ( d − . . . ( d − k + 3) a d X d − k +2 k − e f ( X , . . . , X k − , X k ) , (6)where e f ( X , . . . , X k − , Y ) ∈ L [ X , . . . , X k − , Y ] is a polynomial of de-grees deg X i e f ( X , . . . , X k − , Y ) ≤ d − k + 2 , i = 1 , . . . , k − , and deg Y e f ( X , . . . , X k − , Y ) ≤ d − k + 1 . We write e f ( X , . . . , X k − , X k − + X k ) = X k − h ( X , . . . , X k )+ e f ( X , . . . , X k − , X k )(7)where h ( X , . . . , X k ) ∈ L [ X , . . . , X k ], and taking into account the de-grees above, we getdeg X i h ( X , . . . , X k ) ≤ d − k + 1 , i = 1 , . . . , k − , and deg X i h ( X , . . . , X k ) ≤ d − k, i = k − , k. Taking into account (6) and (7), and using the binomial expansion of( X k − + X k ) d − k +2 , we get∆ X ,...,X k ( f ) = X k − ( d ( d − . . . ( d − k + 3)( d − k + 2) a d X d − k +1 k + e f ( X , . . . , X k )) , where e f ( X , . . . , X k ) ∈ L [ X , . . . , X k ] is defined by e f ( X , . . ., X k ) = h ( X , . . . , X k )+ ( X k − + X k ) d − k +2 − X d − k +2 k − ( d − k + 2) X k − X d − k +1 k X k − , OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 7 and thus satisfy the conditionsdeg X i e f ≤ d − k + 1 , i = 1 , . . . , k − , deg X k e f ≤ d − k. We thus conclude the inductive step. ⊓⊔ If we take k = d in Lemma 1, and then two more consecutive dif-ferences, that is, k = d + 1 and k = d + 2, we obtain the followingconsequence. Corollary 2.
We have ∆ X ,...,X d ( f ) = X d − (cid:16) d ! a d X d + e f ( X , . . . , X d − ) (cid:17) , where e f ( X , . . . , X d − ) ∈ L [ X , . . . , X d − ] is of degrees deg X i e f ( X , . . . , X d − ) ≤ , i = 1 , . . . , d − , and thus ∆ X ,...,X d ,X d +1 ( f ) = d ! a d X d , ∆ X ,...,X d +1 ,X d +2 ( f ) = 0 . Lemma 3.
Let ν ≥ and f = X p ν g ∈ L [ X ] , where g ∈ L [ X ] is ofdegree d < p . Then ∆ X ,...,X d +3 ( f ) = 0 . Proof.
We prove by induction over k ≥ X ,...,X k ( f ) = X k − k X j =1 X p ν j ! g ( X , . . . , X k )+ X k − k − X j =1 X p ν − j g ( X , . . . , X j − , X j +1 , . . . , X k − , X k ) , (8)where g ( X , . . . , X k ) and g ( X , . . . , X j − , X j +1 , . . . , X k − , X k ) are de-fined by (3) and (5).For k = 2, the computations follow exactly as for the general case,so to avoid repetition we prove only the induction step from k − A. OSTAFE k . Using (3), (4) and the induction step, we have∆ X ,...,X k ( f ) = f ( X , . . . , X k − , X k − + X k ) − f ( X , . . . , X k − , X k )= 1 X k − (cid:0) ∆ X ,...,X k − ,X k − + X k ( f ) − ∆ X ,...,X k − ,X k ( f ) (cid:1) = k − X j =1 X p ν j + X p ν k ! g ( X , . . . , X k − , X k − + X k )+ k − X j =1 X p ν − j g ( X , . . . , X j − , X j +1 , . . . , X k − , X k − + X k ) − k − X j =1 X p ν j + X p ν k ! g ( X , . . . , X k − , X k ) − k − X j =1 X p ν − j g ( X , . . . , X j − , X j +1 , . . . , X k − , X k ) . Writing now, as in Lemma 1 (applied to g in place of f ), g ( X , . . . ,X j − , X j +1 , . . . , X k − , X k − + X k )= X k − g ( X , . . . , X j − , X j +1 , . . . , X k )+ g ( X , . . . , X j − , X j +1 , . . . , X k − , X k ) , and the same for g ( X , . . . , X k − , X k − + X k ), and making simple com-putations, we get the equation (8). In fact, using Lemma 1, and thefact that, by (3) we have∆ X ,...,X k ( g ) = X k − g ( X , . . . , X k )and∆ X ,...,X j − ,X j +1 ,...,X k − ,X k ( g )= X k − g ( X , . . . , X j − , X j +1 , . . . , X k − , X k ) , one can rewrite the equation (8) as follows∆ X ,...,X k ( f ) = ∆ X ,...,X k ( g ) k X j =1 X p ν j ! + k − X j =1 X p ν − j ∆ X ,...,X j − ,X j +1 ,...,X k − ,X k ( g )+ X p ν k − g ( X , . . . , X k − , X k ) . OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 9
By Corollary 2 we have∆ X ,...,X d +1 ,X d +2 ( g ) = 0 , ∆ X ,...,X j − ,X j +1 ,...,X d +2 ,X d +3 ( g ) = 0 , and similarly ∆ X ,...,X d +1 ,X d +3 ( g ) = 0, and thus, g ( X , . . . , X d +1 , X d +3 ) = 0 , and thus we conclude the proof. ⊓⊔ As a direct application of Lemma 3, we obtain the following moregeneral result.
Corollary 4.
Let f = X p ν g ν + . . . + X p g + g , g i ∈ L [ X ] , i = 0 . . . , ν, with deg g i + 3 ≤ deg g = d < p, i = 1 . . . , ν, and with g having leading coefficient a d . Then, ∆ X ,...,X d ( f ) = X d − (cid:16) d ! a d X d + e f ( X , . . . , X d − ) (cid:17) , where e f ( X , . . . , X d − ) ∈ L [ X , . . . , X d − ] is of degrees deg X i e f ( X , . . . , X d − ) ≤ ,i = 1 , . . . , d − .Proof. The proof follows directly from Lemma 3. Indeed, we have∆ X ,...,X k ( f ) = ν X i =1 ∆ X ,...,X k (cid:16) X p i g i (cid:17) + ∆ X ,...,X k ( g ) . As d ≥ deg g i + 3, from Lemma 3 we obtain∆ X ,...,X d (cid:16) X p i g i (cid:17) = 0 , and thus, ∆ X ,...,X d ( f ) = ∆ X ,...,X d ( g ) . Applying now Corollary 2, we conclude the proof. ⊓⊔ Remark 5.
We note that when g i , i = 1 , . . . , ν , are constant polyno-mials in Corollary 4, we are in the case f = c ν X p ν + . . . + c X p + g , g i ∈ L [ X ] , c i ∈ L , i = 1 . . . , ν, with ≤ deg g = d < p. Then, we need to take only two differences to eliminate the power of p monomials, that is, we have ∆ X ,X ,X = X g ( X , X , X ) , where g ( X , X , X ) ∈ L [ X , X , X ] with deg X g ( X , X , X ) = d − . Lemma 6.
Let f = X p ν + p ν − + ... + p +1 + g ∈ L [ X ] , where g ∈ L [ X ] is a polynomial of degree ≤ d < p with leadingcoefficient a d . Then ∆ X ,...,X d ( f ) = X d − (cid:16) d ! a d X d + e f ( X , . . . , X d − ) (cid:17) , where e f ( X , . . . , X d − ) ∈ L [ X , . . . , X d − ] is of degrees deg X i e f ( X , . . . , X d − ) ≤ ,i = 1 , . . . , d − .Proof. The case ν = 1 is a special case of Corollary 4. The prooffor ν > p in thedegree and get a polynomial of degree less then p . Indeed, we denotethe monomial m = X p ν + p ν − + ... + p +1 . Simple computations show that, as in Lemma 3, we have∆ X ,X ( m ) = X m ( X , X ) , where m ( X , X ) = X p ν + ... + p + X p ν + ... + p + X X p ν + ... + p − . Next, we have ∆ X ,X ,X ( m ) = X m ( X , X , X ) , where m ( X , X , X ) = X p ν + ... + p − + X p ν + ... + p − . At the next step we already get∆ X ,X ,X ,X ( m ) = 0 . Thus, as d ≥ , we have∆ X ,X ,X ,X ( f ) = ∆ X ,X ,X ,X ( g ) , deg X ∆ X ,X ,X ,X ( g ) = d − ≥ , and the result follows by applying Lemma 1. ⊓⊔ OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 11
Lemma 7.
Let f = g ( l ( x )) ∈ L [ X ] , where g ∈ L [ X ] is a polynomialof degree d < p with leading coefficient a d , and (9) l = ν X i =1 b i X p i ∈ L [ X ] , p ν < q r , is a p -polynomial polynomial. Then ∆ X ,...,X d ( f ) = l ( X d − ) (cid:16) d ! a d l ( X d ) + e f ( l ( X ) , . . . , l ( X d − )) (cid:17) , where e f ( X , . . . , X d − ) ∈ L [ X , . . . , X d − ] is of degrees deg X i e f ( X , . . . , X d − ) ≤ ,i = 1 , . . . , d − .Proof. As the polynomial l is additive, that is l ( X + X ) = l ( X ) + l ( X ), then the proof follows exactly as the proof of Lemma 3 andLemma 6. ⊓⊔ Exponential sums over subspaces
First we introduce the following:
Definition 1.
For < η ≤ , we define a subset A ⊂ L to be η -good if A ∩ b F ) ≤ ( A ) − η for any element b and a proper subfield F of L . The main result of this section follows from the following estimatedue to Bourgain and Glibichuk [4, Theorem 4] which applies to η -goodsets.For 0 < η ≤
1, we define(10) γ η = min (cid:18) , η (cid:19) . Also, all over the paper ψ represents an additive character of L . Lemma 8.
Let ≤ n ≤ . ( r log q ) and A , A , . . . , A n ⊆ L ∗ . Let < η ≤ and γ η be defined by (10) . Suppose A i ≥ , i = 1 , , . . . , n ,and that for every j = 3 , , . . . , n the sets A j are η -good. Assumefurther that A A ( A · · · A n ) γ η > q r (1+ ε ) for some ε > . Then, for sufficiently large q , we have the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A X a ∈ A · · · X a n ∈ A n ψ ( a a . . . a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < A A · · · A n q − . rε/ n . For our results we need an estimate for slightly different (and possiblylarger) sums X a ∈ A · · · X a n ∈ A n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A ψ ( a a . . . a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which we derive directly from Lemma 8. We record this estimate formore general weighted sums, which may be of independent interest forsome future applications. Corollary 9.
Let ≤ n ≤ . ( r log q ) + 1 and A , A , . . . , A n ⊆ L ∗ . Let < η ≤ and γ η be defined by (10) . Suppose A i ≥ , i = 2 , . . . , n , and that for every j = 4 , . . . , n the sets A j are η -good.Assume further that (11) A A ( A · · · A n ) γ η > q r (1+ ε ) for some ε > . Let the weights w i : L → C , i = 1 , . . . , n , be such that (12) X a i ∈ A i | w i ( a i ) | ≤ B i , i = 1 , . . . , n. Then, for sufficiently large q , for the sum J = X a ∈ A · · · X a n ∈ A n w ( a ) . . . w n ( a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A w ( a ) ψ ( a a . . . a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , we have the estimate |J | < n Y i =1 ( B i A i ) / (cid:0) ( A ) − / + 10 q − . rε/ n (cid:1) . Proof.
Squaring and applying the Cauchy-Schwarz inequality, we get |J | ≤ X a ∈ A · · · X a n ∈ A n | w ( a ) | . . . | w n ( a n ) | · X a ∈ A · · · X a n ∈ A n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A w ( a ) ψ ( a a . . . a n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n Y i =2 X a i ∈ A i | w i ( a i ) | ! X a ,b ∈ A w ( a ) w ( b ) · X a ∈ A · · · X a n ∈ A n ψ ( a . . . a n ( a − b )) . OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 13
Applying (12) and Lemma 8 (for the sum over A , . . . , A n ), we obtain |J | ≤ n Y i =2 B / i B / n Y i =2 ( A i ) / +10 n Y i =2 ( A i ) / q − . rε/ n X a ,b ∈ A w ( a ) w ( b ) ! / , where the first summand comes from the case a = b . Taking intoaccount that X a ,b ∈ A w ( a ) w ( b ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A w ( a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A X a ∈ A | w ( a ) | , and using again (12), we conclude the proof. ⊓⊔ Throughout the paper, we slightly abuse this notion of η -good setsand in the case of affine spaces introduce the following: Definition 2.
We say that an affine subspace
A ⊂ L is η -good if A = L + a for some a ∈ L and linear subspace L ⊆ L which is η -goodas in Definition 1. Using Lemma 8 we prove the following estimate of additive charactersums with polynomial argument over an affine subspace of L , which canbe seen as an explicit version of [3, Theorem C]. However, we noticethat all previous such estimates are known for polynomials of degreeless than p . Here we obtain results for more general polynomials.For 0 < ε, η ≤
1, we define(13) δ ( ε, η ) = max (cid:0) , γ − η ( ε − −
1) + 3 (cid:1) . Theorem 10.
Let < ε, η ≤ be arbitrary numbers, γ η and δ ( ε, η ) bedefined by (10) and (13) , respectively. Let A ⊆ L be an η -good affinesubspace of dimension s over K with s ≥ εr. Let d be an integer satisfying the inequalities (14) δ ( ε, η ) ≤ d ≤ min ( p, . log q r ) + 1 , and let f be any polynomial of one of the following forms: (i) f = X p ν g ν + . . . + X p g + g , where g i ∈ L [ X ] , i = 0 . . . , ν, aresuch that deg g = d ≥ deg g i + 3 , i = 1 . . . , ν ; (ii) f = X p ν + p ν − + ... + p +1 + g ∈ L [ X ] , where g ∈ L [ X ] with deg g = d ≥ ; (iii) f = g ( l ( X )) ∈ L [ X ] , where g ∈ L [ X ] with deg g = d and l ∈ L [ X ] is a permutation p -polynomial of the form (9) suchthat l ( L ) is η -good.Let χ : L → C a function satisfying χ ( x + y ) = χ ( x ) χ ( y ) , x, y ∈ L ,and such that X x ∈A | χ ( x ) | d ≤ B. Then, for sufficiently large q , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈A χ ( x ) ψ ( f ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ B ( d +1) / d +1 q s ( − ( d +1) / d +1 ) − rϑ , where (15) ϑ = 0 . ε d . Proof.
Write A = a + L , a ∈ L , where L is a K -linear space of dim K L = s . Making the linear transformation x ∈ L → a + x , we reduce theproblem to estimating the character sum over a linear subspace, S = X x ∈L χ ( x ) ψ ( f ( x )) . We use the method in [24]. For this we square the sum and afterchanging the order of summation and substituting x → x + x , weget | S | = X x ∈L X x ∈L χ ( x ) χ ( x ) ψ ( f ( x ) − f ( x ))= X x ∈L χ ( x ) X x ∈L | χ ( x ) | ψ (∆ x ,x ( f )) , where ∆ X ,X ( f ) is defined by (3).Squaring and applying the Cauchy-Schwarz inequality again, we get | S | ≤ q s X x ∈L | χ ( x ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈L | χ ( x ) | ψ (∆ x ,x ( f )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = q s X x ∈L | χ ( x ) | X x ,x ∈L | χ ( x ) | | χ ( x ) | ψ (∆ x ,x ( f ) − ∆ x ,x ( f )) . Substituting x → x + x , we get | S | ≤ q s X x ,x ∈L | χ ( x ) | | χ ( x ) | X x ∈L | χ ( x ) | ψ ( x ∆ x ,x ,x ( f )) , OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 15 where ∆ X ,X ,X ( f ) is defined by (3).Simple inductive argument shows that applying this procedure, thatis, squaring and applying the Cauchy-Schwarz inequality, d − f correspondingto (i) and (ii), we get to the exponential sum | S | d − ≤ q s (2 d − − d ) X x ,x ,...,x d − ∈L | χ ( x ) | d − | χ ( x ) | d − . . . | χ ( x d − ) | d − · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x d ∈L | χ ( x d ) | d − ψ ( d ! a d x x . . . x d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We apply now Corollary 9 with4 ≤ n = d ≤ . log q r + 1and A = . . . = A n = L . We note that the condition (14) implies that s (2 + ( d − γ η ) > r ( ε + 1) , and thus the condition (11) in Corollary 9 is also satisfied. Moreover,we have X x ∈L | χ ( x ) | d − ! ≤ q s X x ∈L | χ ( x ) | d ≤ Bq s , and thus X x ∈L | χ ( x ) | d − ≤ B / q s/ . We get | S | d − ≤ B ( d +1) / q s ( d − − ( d +1) / ) − . rε/ d . which immediately implies the result.For the case (iii), proceeding the same but applying Lemma 7 andtaking into account that l is a permutation polynomial, we get to theexponential sum | S | d − ≤ q s (2 d − − d ) X x ,x ,...,x d − ∈L | χ ( x ) | d − χ ( x ) | d − . . . | χ ( x d − ) | d − · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x d ∈L | χ ( x d ) | d − ψ ( d ! a d l ( x ) l ( x ) . . . l ( x d )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = q s (2 d − − d ) X x ,x ,...,x d − ∈ l ( L ) | χ ( l − ( x )) | d − χ ( l − ( x )) | d − . . . · | χ ( l − ( x d − )) | d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x d ∈ l ( L ) | χ ( l − ( x d )) | d − ψ ( d ! a d x x . . . x d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where l ( L ) is a subset of L of cardinality q s (we note that if l is a q -polynomial, then l ( L ) is actually a K -linear subspace of L ) and l − is the compositional inverse of l , which is again a linearised polynomialof the form (9), see [29, Theorem 4.8]. Thus, we have that χ (cid:0) l − ( x + y ) (cid:1) = χ (cid:0) l − ( x ) + l − ( y ) (cid:1) = χ (cid:0) l − ( x ) (cid:1) χ (cid:0) l − ( y ) (cid:1) and X x ∈ l ( L ) | χ (cid:0) l − ( x ) (cid:1) | d = X x ∈L | χ ( x ) | d ≤ B. As we also assume that l ( L ) is η -good, the estimate follows the sameby applying Corollary 9. ⊓⊔ Remark 11.
We note that, by [21, Theorem 7.9] , a p -polynomial l ∈ L [ X ] as defined by (9) is a permutation polynomial if and only if it hasonly the root in L . Remark 12.
In Theorem 10, (iii), we assume that the set l ( L ) is η -good. We note that when l ( X ) = X p ν , then l ( b F ) = b p ν F , for anysubfield F of L and any element b not in F , and thus l ( L ) is η -good.Another immediate example can be given for a prime q = p and also aprime r . Since the only proper subfield F of L is F p , if s ≥ , that is, L ≥ p then, l ( L ) ∩ b F ) ≤ p ≤ l ( L ) / . Remark 13.
Probably the most natural examples of the function χ ( x ) in Theorem 10 is given by exponential functions such as χ ( x ) = exp π m X j =1 ζ j Tr L | F p ( τ j x ) ! for some ζ j ∈ R and τ j ∈ L . Values of polynomials in subspaces
In this section we give upper bounds for I f ( A , B ) defined by (1),that is, the cardinality of f ( A ) ∩ B , for a polynomial f ∈ L [ X ] andaffine subspaces A , B of L over K .For our first result we use the Weil bound, see [21, Theorem 5.38],in a standard way. We recall it for the sake of completeness and use itas a benchmark for further improvements. Lemma 14.
Let f ∈ L [ X ] be of degree d ≥ with ( d, p ) = 1 , and let ψ be a nontrivial additive character of L . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X c ∈ L ψ ( f ( c )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( d − q r/ . OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 17
Theorem 15.
Let f ∈ L [ X ] be a polynomial of degree d ≥ , ( d, p ) = 1 , A ⊆ L and B ⊆ L affine subspaces of dimension s and m , respectively,over K . Then, we have I f ( A , B ) = q s + m − r + O (cid:0) dq r/ (cid:1) . Proof.
As in Theorem 10, we can reduce the problem to estimating I f ( L , L ), where L , L are linear spaces.Let β , . . . , β r − s be the basis for the complementary space of L and ω , . . . , ω r − m be the basis for the complementary space of L , that is, u ∈ L and v ∈ L if and only if(16)Tr L | K ( β i u ) = 0 , i = 1 , . . . , r − s, Tr L | K ( ω i v ) = 0 , i = 1 , . . . , r − m. We use the relations (16) to give an upper bound for I f ( A , B ) = I f ( L , L ). Indeed, let ψ be a nontrivial additive character of L . Usingadditive character sums to count the elements u ∈ L such that theelements of f ( u ) satisfy (16), we have I f ( L ) = 1 q r − s − m X x ∈ L X c i ,d j ∈ K i =1 ,...,r − sj =1 ,...,r − m ψ r − s X i =1 c i β i x + r − m X j =1 d j ω j f ( x ) ! = q s + m − r + 1 q r − s − m X c i ,d i ∈ K i =1 ,...,r − sj =1 ,...,r − m ∗ X x ∈ L ψ r − s X i =1 c i β i x + r − m X j =1 d j ω j f ( x ) ! , where the first term is given by c i = d j = 0, i = 1 , . . . , r − s , j =1 , . . . , r − m , and P ∗ means that at least one element c i , d j = 0.We notice that since deg f = d ≥
2, nontrivial linear combinations r − s X i =1 c i β i x + r − m X j =1 d j ω j f ( x ) , c i , d j ∈ K , i = 1 , . . . , r − s, j = 1 , . . . , r − m, that appear in the inner sum are all nonconstant polynomials. Indeed,assume that this is not the case, and without loss of generality wecan also assume that d i = 0 for at least one i = 1 , . . . , t . Then, thevanishing of the leading coefficients (of the monomial X d ) r − m X i =1 d i ω i X d = 0implies that the elements ω , . . . , ω r − m are linearly dependent as ele-ments of L seen as a vector space over K , which contradicts the hy-pothesis. We can apply now the Weil bound given by Lemma 14 to the sumover x ∈ L and conclude the proof. ⊓⊔ We note that the bound of Theorem 15 is nontrivial whenever dq r/ r (1 / ε ), for some ε > s and m . We recall first a similar result that was recentlyobtained in [22, Theorem 7] for the case A = B and only for polyno-mials of degree smaller than p .Let f ∈ L [ X ] be of degree d = deg f with p > d ≥ A ⊆ L be an affine subspace of dimension s over K such that for any subfield F ⊆ L and any b ∈ L we have L ∩ b F ) ≤ max (cid:26) ( L ) / , q s (1 − ρ d ) (cid:27) , where A = a + L for some a ∈ F and a linear subspace L ⊆ L . Thenthe following estimate is obtained in [22, Theorem 7]:(17) I f ( A , A ) ≪ q s (1 − κ d ) , with η d = 4277 · d − − , κ d = 4277 · d − + 3 , and for d ≥ ϑ d = η d + ϑ d − − η d ϑ d − , ρ d = η d + ϑ d − η d ϑ d , where η = ϑ = 1 / Theorem 16.
Let < ε, η ≤ be arbitrary numbers and let γ η and δ ( ε, η ) be defined by (10) and (13) , respectively. Let A ⊆ L be an η -good affine subspace of dimension s over K with s ≥ εr, and B ⊆ L another affine subspace of dimension m over K . Let d and f be as in Theorem 10. Then (cid:12)(cid:12) I f ( A , B ) − q s + m − r (cid:12)(cid:12) ≤ q s − rϑ , where ϑ is defined by (15) . OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 19
Proof.
As in the proof of Theorem 15 (where we take the sum over L ,not all the field L ), we have I f ( A , B ) = 1 q r − m X c i ∈ K i =1 ,...,r − m X x ∈L ψ r − m X i =1 c i ω i f ( x ) ! , = q s + m − r + 1 q r − m X c i ∈ K i =1 ,...,r − m ∗ X x ∈L ψ r − m X i =1 c i ω i f ( x ) ! , where the first term corresponds to c i = 0 for all i = 1 , . . . , r − m and P ∗ means that at least one c i = 0. We denote T = X x ∈L ψ r − m X i =1 c i ω i f ( x ) ! . We apply Theorem 10 (with χ ( x ) = 1, x ∈ L , and B = q s ) for thesum T with the polynomial F = r − m X i =1 c i ω i f ( X ) ∈ L [ X ] , which is of degree d as at least one c i = 0. We get | T | ≤ q s − rϑ , and thus we conclude the proof. ⊓⊔ We note that I f ( A , B ) > m > r (1 − ϑ ). Fur-thermore, for any fixed ρ > − ϑ and m ≥ rρ , Theorem 16 gives anasymptotic formula for I f ( A , B ) as q r → ∞ .When A = B in Theorem 16, we get the estimate I f ( A , A ) ≤ q s − rϑ . This bound improves the estimate (17) obtained in [22] for s < . (cid:18) (cid:19) d rε. In particular, if ε = s/r , it always improves (17) whenever d satisfies thecondition (14). Moreover, Theorem 16 generalises (17) as this estimatewas obtained in [22] only for polynomials of degree d < p .Note also that the results of Roche-Newton and Shparlinski [22] al-ways required η ≥ / Corollary 17.
If under the conditions of Theorem 16 we have f ( A ) ⊆B , then B = L . Proof.
Indeed, if f ( A ) ⊆ B , then from Theorem 16 we derive q s = I f ( A , B ) ≤ q s + m − r + 2 q s − rϑ , which is possible only if m = r . ⊓⊔ Theorem 16 has also direct consequences on the image and kernelsubspaces of q -polynomials defined by(18) l = ν X i =1 b i X q i ∈ L [ X ] , ν < r. Then, for an affine subspace B of L of dimension m ≤ r , the image set l ( B ) = { l ( x ) | x ∈ B} is a K -affine subspace of dimension at most m .Moreover, we denote by Ker ( l ) the set of zeroes of the polynomial l .By [21, Theorem 3.50], Ker ( l ) is a K -linear subspace of F q t , where F q t is the field extension of L containing all the roots of l . Taking now thetrace over L , we have that Tr F qt | L ( Ker ( l )) is a K -linear subspace of L .Under the conditions of Theorem 16, for any q -polynomial l ∈ L [ X ]defined by (18), we have (cid:12)(cid:12) I f ( A , l ( B )) − q s + m − r (cid:12)(cid:12) ≤ q s − rϑ , where ϑ is defined by (15). The same estimate holds for I f ( A , Tr F qt | L ( Ker ( l )))with m replaced with dim K Tr F qt | L ( Ker ( l )).Moreover, as in Corollary 17, we see that f ( A ) is not included in l ( B ) for any proper subspace B ⊆ L or in Tr F qt | L ( Ker ( l )).It would be certainly interesting to find upper bounds for the in-tersection of image sets of polynomials on affine subspaces. That is,given f, g ∈ L [ X ], find estimates for the size of f ( A ) ∩ g ( A ) for a givenproper affine subspace A ⊂ L . For prime fields, Chang shows in [7] thatthe intersection of the images of two polynomials on a given interval issparse. In the case of arbitrary finite fields, several such estimates aregiven in [10] for very special classes of polynomials and affine spaces.5. Polynomial orbits in subspaces
As in [22], one can obtain immediately from Theorem 16 the follow-ing consequence about the number of consecutive iterates falling in asubspace. We recall that for a polynomial f ∈ L [ X ] and element u ∈ L ,we define T f,u = f ( u ) as defined by (2). OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 21
Corollary 18.
Let < ε, η ≤ be arbitrary numbers and let γ η and δ ( ε, η ) be defined by (10) and (13) , respectively. Let A ⊆ L be an η -good affine subspace of dimension s over K with s ≥ εr. Let d and and f be as in Theorem 10. If for some u ∈ L and an integer N with ≤ N ≤ T f,u we have f ( n ) ( u ) ∈ A , n = 0 , . . . , N − , then q s ≥ N q rϑ , where ϑ is defined by (15) .Proof. The result follows directly from Theorem 16 as N ≤ I f ( A , A ) ≤ q s − rϑ . ⊓⊔ Remark 19.
Similarly to Corollary 18 (replacing A with the imagespace of a linearised polynomial l ), based on the discussion after Corol-lary 17, one can obtain estimates for the number of consecutive ele-ments in the orbit of a polynomial of the form defined in Theorem 10,that fall in the orbit of l in any point of L . We also note that the proof of [22, Theorem 6], using Theorem 16,can give information about the number of arbitrary (not necessarilyconsecutive) iterates falling in a subspace. For the sake of completenesswe repeat the argument of [22, Theorem 6] for the case of subspacesinstead of subfields for which this result has been obtained.We present our bounds in terms of the the parameter ρ which is afrequency of iterates of f ∈ L [ X ] in an affine space, that is, ρ = M/N ,where M is the number of positive integers n ≤ N with f ( n ) ( u ) ∈ A .Again, we obtain a power improvement over the trivial bound q s ≥ ρN (where s = dim A ). Theorem 20.
Let < ε, η ≤ and let γ η and δ ( ε, η ) be defined by (10) and (13) , respectively. Let A ⊆ L be an η -good affine subspace ofdimension s ≥ εr over K . Let f ∈ L [ X ] be a polynomial of degree d such that for N ≤ T f,u we have f ( n ) ( u ) ∈ A for at least ρN ≥ valuesof n = 1 , . . . , N . If δ ( ε, η ) ≤ d ρ − ≤ min ( p, . log q r ) + 1 , then q s ≥ ρ N q rϑ ρ , where (19) ϑ ρ = 0 . ε d /ρ . Proof.
We follow exactly the same proof as in [22, Theorem 6]. Let 1 ≤ n < . . . < n M ≤ N be all values such that f ( n i ) ( u ) ∈ A , i = 1 , . . . , M .We denote by A ( h ) the number of i = 1 , . . . , M − n i +1 − n i = h .Clearly N X h =1 A ( h ) = M − N X h =1 A ( h ) h = n M − n ≤ N. Thus, for any integer H ≥ H X h =1 A ( h ) = M − − N X h = H +1 A ( h ) ≥ M − − ( H + 1) − N X h = H +1 A ( h ) h ≥ M − − ( H + 1) − N. Hence there exists k ∈ { , . . . , H } with(20) A ( k ) ≥ H − (cid:0) M − − ( H + 1) − N (cid:1) . Let H = ⌊ ρ − ⌋ ≥
1. Then H − (cid:0) M − − ( H + 1) − N (cid:1) ≥ M − H ≥ ( M − N and we derive from (20) that(21) A ( k ) ≥ ( M − N = ρ N (cid:18) − M (cid:19) ≥ ρ N . Let J be the set of j ∈ { , . . . , M − } with n j +1 − n j = k . Then wehave f ( n j ) ( u ) ∈ A and f ( n j +1 ) ( u ) = f ( k ) (cid:0) f ( n j ) ( u ) (cid:1) ∈ A , that is (cid:0) f ( n j ) ( u ) , f ( k ) (cid:0) f ( n j ) ( u ) (cid:1)(cid:1) ∈ A ∩ f ( k ) ( A ) . Thus, A ( k ) ≤ I f ( k ) ( A , A ), and from (21) and Theorem 16, we get ρ N ≤ q s − rϑ ρ , where ϑ ρ is defined by (19). We thus conclude the proof. ⊓⊔ OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 23
One can also obtain information on the intersection of orbits of apolynomial f of degree d < p with orbits of a q -polynomial l (see alsothe discussion after Corollary 17). Corollary 21.
Let < ε, η ≤ and let γ η and δ ( ε, η ) be defined by (10) and (13) , respectively. Let f ∈ L [ X ] be a polynomial of degree d and l ∈ L [ X ] a linearsied polynomial of the form (18) such that l ( L ) is an η -good linear subspace of dimension s ≥ εr over K . Let M = f ( u ) ∩ Orb l ( v )) , and ρ = M/ min( T f,u , T l,v ) the frequency of intersection of the orbits.If δ ( ε, η ) ≤ d ρ − ≤ min ( p, . log q r ) + 1 , then q s ≥ ρ min( T f,u , T l,v )32 q rϑ ρ , where ϑ ρ is defined by (19) .Proof. As Orb l ( v ) ⊂ l ( L ), the proof follows exactly as the proof ofTheorem 20, but with A replaced with l ( L ) and N replaced withmin( T f,u , T l,v ). ⊓⊔ Exponential sums over consecutive integers
In this section we consider q = p . For a positive integer n ≤ p r − p -adic representation(22) n = n + n p + . . . + n s − p s − , ≤ n j < p, j = 0 , . . . , s − , for some s ≤ r .In this section we fix a basis ω , . . . , ω r − of L over F p and define(23) ξ n = s − X j =0 n j ω j . Let 1 ≤ N ≤ p r − f ∈ L [ X ] a polynomial of degree d and ψ anadditive character of L . In this section we estimate the exponentialsum S ( N ) = X n ≤ N χ ( n ) ψ ( f ( ξ n )) , where χ : N → C is a p -multiplicative function, that is, it satisfies thecondition χ (cid:0) m + tp k (cid:1) = χ ( m ) χ (cid:0) tp k (cid:1) for all k ≥ t ≥ ≤ m < p k . This class of functions, as wellas the closely related class of p -additive functions have been studied in classical works of Gelfond [16] and Delange [14], see also [15, 17, 20]and references therein for more recent developments.A large family of such function can be obtained as(24) χ ( n ) = exp πi s − X j =0 α j n j ! , where α j , j = 0 , , . . . , is a fixed infinite sequence of real numbers and n is given by the p -adic representation as in (22), see also [19] for amore general class. In particular taking α j = αp j and α j = α for a real α , we obtain the following two natural examples, χ ( n ) = exp (2 πiαn ) and χ ( n ) = exp (2 πiασ p ( n )) , respectively, where σ p ( n ) is the sum of p -ary digits of n .For simplicity we consider the family (24) in the next result. Theorem 22.
Let < ε, η ≤ be arbitrary numbers and let γ η and δ ( ε, η ) be defined by (10) and (13) , respectively. Let p s − ≤ N ≤ p s − for some s ≤ r satisfying s ≥ εr, and assume the linear subspace L s ⊆ L spanned by ω , . . . , ω s − is η -good. Let f ∈ L [ X ] be a polynomial of the form (i), (ii) or (iii) asdefined in Theorem 10 with d satisfying the condition (25) δ ( ε/ , η/
2) + 1 ≤ d ≤ min ( p, . log p r ) + 2 , and ψ an additive character of L . Let χ : N → C be a p -multiplicativefunction defined by (24) . Then | S ( N ) | ≤ ( N p ) − η/ + 2 N p − rϑ η / , where (26) ϑ η = 0 . ε (1 − η/ d − . Proof.
Let K = ⌈ s (1 − η/ ⌉ and M = p K ⌊ N/p K ⌋ −
1. Our sumbecomes(27) | S ( N ) | ≤ | S ( M ) | + p K ≤ | S ( M ) | + p ( N p ) − η/ . From the definition of M , we have that M = K − X i =0 ( p − p i + T p K , for some T ≤ p s − K − − ≤ N p − K − OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 25
We have | S ( M ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m
2. As m runs over the interval [0 , p K − ξ m runs over all theelements of L K , and moreover, ζ t = ζ t for t = t .Our sum becomes | S ( M ) | ≤ p K X t ,t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈L K ψ ( f ( x + ζ t ) − f ( x + ζ t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N p K + p K X t ,t ≤ T,t = t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈L K ψ ( F t ,t ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where F t ,t ( X ) = f ( X + ζ t ) − f ( X + ζ t ) ∈ L [ X ].We note that, as f ∈ L [ X ] is a polynomial of the form (i), (ii) or(iii) as defined in Theorem 10, then F t ,t is a non constant polynomialof the same form as f . When f is of the form (i), we have f = X p ν g ν + . . . + X p g + g , where g i ∈ L [ X ] , i = 0 . . . , ν, are such that deg g = d ≥ deg g i + 3, i = 1 . . . , ν . Then we get F t ,t ( X ) = X p ν ( g ν ( X + ζ t ) − g ν ( X + ζ t )) + . . . + X p ( g ( X + ζ t ) − g ( X + ζ t )) + F ,t ,t ( X ) , where F ,t ,t ( X ) = g ( X + ζ t ) − g ( X + ζ t )+ ν X i =1 (cid:16) ζ p i t g i ( X + ζ t ) − ζ p i t g i ( X + ζ t ) (cid:17) . For t = t , we note that g i ( X + ζ t ) − g i ( X + ζ t ), i = 0 , . . . , ν , is anonconstant polynomial of degree equal to deg g i −
1, and d − F ,t ,t ≥ deg ( g i ( X + ζ t ) − g i ( X + ζ t )) + 3 . Thus, F t ,t is of the same form and satisfies the same conditions as f .Similarly, if f is of the form (ii) of Theorem 10, that is f = X p ν + p ν − + ... + p +1 + g ∈ L [ X ] , where g ∈ L [ X ] with deg g = d ≥
5, then F t ,t ( X ) = g ( X + ζ t ) − g ( X + ζ t )is a non constant polynomial of degree d − ≥ f is of the form (iii) of Theorem 10, that is, f = g ( l ( x )) withdeg g = d and some permutation p -polynomial l ∈ L [ X ], then F t ,t ( X ) = g ( l ( X ) + l ( ζ t )) − g ( l ( X ) + l ( ζ t )) = G t ,t ( l ( X )) , where G t ,t ( X ) = g ( X + l ( ζ t )) − g ( X + l ( ζ t )) ∈ L [ X ] is of degree d − s ≥ εr , then K ≥ s (1 − η/ ≥ ε η r , where ε η = ε (1 − η/
2) by thehypothesis. Since K ≥ s (1 − η/ > s − η − η/ F of L , L K ∩ b F ) ≤ L s ∩ b F ) ≤ p s (1 − η ) < p K (1 − η/ . Moreover, from condition (25), we have d − ≥ δ ( ε/ , η/ ≥ δ ( ε η , η/ d replaced by d − ε replaced by ε η and η replaced by η/ | S ( M ) | ≤ N p K + 2 p K T p K − rϑ η ≤ N p s (1 − η/ + 2 N p − rϑ η ≤ ( N p ) − η/ + 2 N p − rϑ η , OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 27 where ϑ η is given by (26) and thus, recalling (27), we conclude theproof. ⊓⊔ We also note that we have not put any efforts in optimising the con-dition (25) in Theorem 22. For example, if one imposes the condition δ ( ε (1 − . η ) , . η ) + 1 ≤ d ≤ min ( p, . log p r ) + 2 , then one obtains the slightly better bound | S ( N ) | ≤ ( N p ) − . η + 2 N p − rϑ η / , where ϑ η = 0 . ε (1 − . η )2 d − . Remark 23.
We note that similarly to the proof of Theorem 22 wecan derive directly from Theorem 10 a bound for the exponential sum R ( N ) = X n ≤ N χ ( ξ n ) ψ ( f ( ξ n )) , where f ∈ L [ X ] is of the form (i), (ii) or (iii) as defined in Theorem 10with d satisfying the condition δ ( ε/ , η/ ≤ d ≤ min ( p, . log p r ) + 1 . Let χ : L → C satisfy the conditions χ ( x + y ) = χ ( x ) χ ( y ) , x, y ∈ L ,and X x ∈A s | χ ( x ) | d ≤ B. Then, one obtains | R ( N ) | ≤ B ( d +1) / d +1 N p − K ( d +1) / d +1 − rϑ η + p ( N p ) − η/ max n ≤ N | χ ( ξ n ) | , where ϑ η = 0 . ε (1 − η/ d . Indeed, as in the proof of Theorem 22 we reduce the problem to es-timating | R ( M ) | , where M = p K ⌊ N/p K ⌋ − . As in the proof of The-orem 22, the set of integers n ≤ M is of the form (28) , and thus, wenow see from (23) that the set of ξ n is partitioned into the union of T + 1 affine spaces of the shape A ( t ) = L K + ζ t , where L K is the K -dimensional linear subspace defined by the basis elements ω , . . . , ω K − of L over F p , and with some ζ t ∈ L , ≤ t ≤ T . As there are at most
N/q K elements ξ t ∈ L corresponding to t ≤ T as discussed above, our sum becomes | R ( M ) | ≤ N q − K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈A ( t ) χ ( x ) ψ ( f ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where A ( t ) = L K + ζ t for some ζ t ∈ L , ≤ t ≤ T . Now, the estimatefollows applying Theorem 10 to the sum R ( M ) .Moreover, if N = p s − , for some s ≤ r , the set of elements ξ n corresponding to n ≤ N given by (23) defines an affine subspace A of L of dimension s . This case is exactly Theorem 10, and thus | R ( N ) | ≤ B ( d +1) / d +1 p s ( − ( d +1) / d +1 ) − rϑ , where ϑ is defined by (15) (but with d replaced by d − ). Waring problem in intervals and subspaces
Let f ∈ L [ X ] be a polynomial of degree d . In this section we considerfirst the Waring problem over an affine subspace A of L of dimension s , that is the question of the existence and estimation of a positiveinteger k such that, for any y ∈ L , the equation(29) f ( x ) + . . . + f ( x k ) = y, is solvable in x , . . . , x k ∈ A .In particular, we denote by g ( f, q, s ) the smallest possible value of k in (29) and put g ( f, q, s ) = ∞ if such k does not exist.We obtain the following direct consequence of Theorem 10. Theorem 24.
Let < ε, η ≤ be arbitrary numbers and let γ η and δ ( ε, η ) be defined by (10) and (13) , respectively. Let A ⊆ L be an η -good affine subspace of dimension s over K with s ≥ εr. If f ∈ L [ X ] is a polynomial of the form (i), (ii) or (iii) as defined inTheorem 10, then for k ≥ with (cid:18) q rϑ (cid:19) k − > Dq r − s , where ϑ is defined by (15) and D = deg f in the cases (i) and (ii) and D = deg g in the case (iii), we have g ( f, q, s ) ≤ k. OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 29
Proof.
We use again exponential sums to count the number of solutions N k of the equation (29), that is, N k = 1 q r X u ∈ L X x ,...,x k ∈A ψ u k X i =1 f ( x i ) − y !! and thus | N k − q sk − r | ≤ q r X u ∈ L ∗ | S u | k = 1 q r X u ∈ L ∗ | S u | k − | S u | ≤ q r X u ∈ L ∗ | S u | k − X x ,x ∈A ψ ( u ( f ( x ) − f ( x ))) , where S u = X x ∈A ψ ( uf ( x )) . Using Theorem 10 for the sum S u and the estimate Dq s (for fixed x ∈ A , there are at most D zeros of f ( x ) − f ( X )) for the inner sum,we obtain (cid:12)(cid:12) N k − q sk − r (cid:12)(cid:12) ≤ k − Dq s ( k − − rϑ ( k − , where ϑ is defined by (15). Imposing now N k >
0, we conclude theproof.The statement for the polynomial of the type (iii) in Theorem 10follows as l is a permutation p -polynomial as defined in Theorem 10. ⊓⊔ If D is fixed in Theorem 24, then for k > r − sr ϑ − + 2and sufficiently large q s we have g ( f, q, s ) < k .Next we consider q = p , and for an integer n ≤ N , we have ξ n definedby (23). We also study the question of the existence of a positive integer k such that for any y ∈ L , the equation f ( ξ n ) + . . . + f ( ξ n k ) = y is solvable in positive integers n , . . . , n k ≤ N . As above, we denote by G ( f, p, N ) the smallest such value of k and put G ( f, p, N ) = ∞ if such k does not exist. Corollary 25.
Let f ∈ L [ X ] be a polynomial of the form (i), (ii) or(iii) as defined in Theorem 10 and p s − ≤ N < p s for some s ≤ r satisfying s ≥ εr, and assume the linear subspace L s ⊆ L spanned by ω , . . . , ω s − is η -good. Then for k ≥ with (cid:18) p rϑ (cid:19) k − > Dp r − s +1 , where ϑ is defined by (15) and D = deg f in the cases (i) and (ii) and D = deg g in the case (iii), we have G ( f, p, N ) ≤ k. Proof. As N ≥ p s − , we have that G ( f, p, N ) ≤ g ( f, p, s − s replaced with s −
1, and with q replaced by p . ⊓⊔ We note that Corollary 25 follows also by applying directly Theo-rems 22, however the estimate obtained would be slightly weaker.8.
Remarks and open questions
We note that we could prove Theorem 20 only for polynomials ofdegree less than p . The reason behind this is that when one iteratesthe polynomial f of the form (i), (ii) or (iii), the shape changes andthus we cannot apply anymore Theorem 10. It would be interesting toextend such a result for more general polynomials.Theorem 16 can also be translated into the language of affine dis-persers, see [1]. We consider q = p prime and L = F p r . Definition 3.
A function f : L → F p is an F p -affine disperser fordimension s if for every affine subspace A of L of dimension at least s , we have f ( A ) > . As a direct consequence of Theorems 16, we obtain the followingresult.
Corollary 26.
Let < ε, η ≤ and let f ∈ L [ X ] be a polynomial asdefined in (i), (ii) or (iii) of Theorem 16. Then π ( f ) , where π : L → F p is a nontrivial F p -linear map, is an affine disperser for dimensiongreater than εr . We note that condition (14) shows that the larger ε is, the smallerthe degree d is, where d is defined as in Theorem 16. For example, if ε = 12 and η ≤ d > δ (1 / , η ) = γ − η + 3 = 156453. Furthermore, if ε = 13 and η ≤ OLYNOMIAL VALUES IN AFFINE SUBSPACES OF FINITE FIELDS 31 then d > δ (1 / , η ) = 2 γ − η + 3 = 312903.As mentioned in [22], obtaining analogues of Theorem 10, and thusof the rest of results of this paper, for rational functions is an importantopen direction. For this one has to obtain estimates for the exponentialsum S = X x ∈L ψ ( h ( x )) , where h ∈ L ( X ) is a rational function and ψ a nontrivial additivecharacter. Even the case h ( X ) = X − is still open.Also of interest is obtaining estimates for S = X x ∈G ψ ( h ( x )) , where G is a multiplicative subgroup of L ∗ . We note that for the primefield case, such a result would follow from [2, Theorem 1].Of interest is also the multivariate case of Tehorem 16, that is, given F ∈ L [ X , . . . , X n ] and A , . . . , A n , B affine subspaces of L , estimatethe size of F ( A , . . . , A n ) ∩ B . Acknowledgements
The author would like to thank Igor Shparlinski for suggesting someextensions of initial results, and for his important comments on earlierversions of the paper. The author is also grateful to the Max PlanckInstitute for Mathematics for hosting the author for two months duringthe program “Dynamics and Numbers” when important progress onthis paper was made.During the preparation of this paper the author was supported bythe UNSW Vice Chancellor’s Fellowship.
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