Counting polynomial subset sums

Canadian Journal of Mathematics, 1969

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm ] for the finite field with pm elements and [pm ]′ for [pm ] – {0}. We consider in this paper only subsets H of [pm ] for which |H| = h satisfies 1.1 If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm ]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p 0(d), for some prime p 0, depending only on d, which is greater than d.

Journal of Combinatorial Theory, Series A, 2014

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 ,. .. , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts that V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d d−s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q-rational points is established.

Acta Arithmetica, 2014

2017

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known. We consider a class $\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.

Finite Fields and Their Applications, 1996

2017

A classical result on value sets of non-permutation polynomials over finite fields is due to Wan (1993). Denoting the cardinality of the value set of f 2 Fq[x] by jVf j, Wan's result gives the upper bound JVx, where d is the degree of f. A proof of this bound due to Turnwald, which was obtained by the use of symmetric polynomials is given in Chapter 2. A generalization of this result was obtained by Aitken that we also describe here. The work of Aitken focuses on value sets of pairs of polynomials in Fq[x], in particular, he studies the size of the intersection of their value sets. We present pairs of particular polynomials whose value sets do not only have the same size but are actually identical. Clearly, a permutation polynomial f of Fq[x] satisfies jVf j = q. In Chapter 3, we discuss permutation behaviour of pairs of polynomials in Fq[x].

Finite Fields and Their Applications, 2011

In this paper we study the relation between coefficients of a polynomial over finite field F q and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m | q − 1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m q−1 and m ≤ q−1 2. As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q 3/2) operations.

Finite Fields and Their Applications, 2022

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.

Journal of the London Mathematical Society, 2006

Our first result is a 'sum-product' theorem for subsets A of the finite field Fp, p prime, providing a lower bound on max(|A + A|, |A · A|). The second and main result provides new bounds on exponential sums

Proceedings of the American Mathematical Society, 1993

Let ¥q be the finite field of q elements, and let Vf be the number of values taken by a polynomial f{x) over ¥q. We establish a lower bound and an upper bound of Vf in terms of certain invariants of f(x). These bounds improve and generalize some of the previously known bounds of Vf. In particular, the classical Hermite-Dickson criterion is improved. Our bounds also give a new proof of a recent theorem of Evans, Greene, and Niederreiter. Finally, we give some examples which show that our bounds are sharp.

In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.

Moscow Mathematical Journal

We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-rational points on a projective algebraic variety defined by r linearly independent homogeneous polynomial equations of degree d in m + 1 variables with coefficients in the finite field Fq with q elements, when d < q. It is shown that this formula holds in the affirmative for several values of r. In the general case, we give explicit lower and upper bounds for er(d, m) and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindström Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.

Given a finite subgroup $G$ of the group of units of a commutative unital ring $R$ and a multivariate polynomial $f$ in $R[X_1,\ldots,X_k]$, we evaluate the sum of the $f(x_1,\ldots,x_k)$ for all choices of pairwise distinct $x_1,\ldots,x_k$ in $G$ whenever the subgroup $G$ satisfies a minimax constraint, which always holds if $R$ is a field. In particular, let $p^m$ be a power of an odd prime, $n$ a positive integer, and $a_1,\ldots,a_k$ integers with sum divisible by $\varphi(p^m)$ such that $\mathrm{gcd}(a_{i_1}+\cdots+a_{i_j},p(p-1))$ is smaller than $(p-1)/\mathrm{gcd}(n,\varphi(p^m))$ for all non-empty proper subsets $\{i_1,\ldots,i_j\}$ of $\{1,\ldots,k\}$; then the following congruence holds $$\sum x_1^{a_1}\cdots x_k^{a_k} \equiv \frac{\varphi(p^m)}{\mathrm{gcd}(n,\varphi(p^m))}(-1)^{k-1}(k-1)! \hspace{1mm}\pmod{p^m}, %(-1)^{k}(k-1)!p^{m-1} \bmod{p^m},$$ where the summation is taken over all pairwise distinct $1\le x_1,\ldots,x_k\le p^m$ such that each $x_i$ is a $n$-th res...

Contemporary Mathematics

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).

Discrete Mathematics, 1999