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Igor E. Shparlinski

Publications and source records attributed to Igor E. Shparlinski.

At least 181 records · Page 10Linked to original sources
arXiv2014

On the Density of Integer Points on the Generalised Markoff-Hurwitz and Dwork Hypersurfaces

We use bounds of mixed character sums modulo a prime $p$ to estimate the density of integer points on the hypersurface $$ f_1(x_1) + \ldots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} $$ for some polynomials $f_i \in {\mathbb Z}[X]$, nonzero integer $a$ and positive integers $k_i$ $i=1, \ldots, n$. In the case of $$ f_1(X) = \ldots = f_n(X) = X^2 \quad \text{and}\quad k_1 = \ldots = k_n =1 $$ the above congruence is known as the Markoff-Hurwitz hypersurface, while for $$ f_1(X) = \ldots = f_n(X) = X^n\quad \text{and}\quad k_1 = \ldots = k_n =1 $$ it is known as the Dwork hypersurface. Our result is substantially stronger than those known for general hypersurfaces.

math.NT↗
arXiv2014

On the singularity of the Demjanenko matrix of quotients of Fermat curves

Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal C}_{k,\ell})$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell}$ is singular (equivalently, for which the rank of the Hodge group of ${\mathrm Jac}({\mathcal C}_{k,\ell})$ is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such $k$ when $\ell$ tends to infinity.

math.NT↗
arXiv2014

Counting Additive Decompositions of Quadratic Residues in Finite Fields

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S = \{a+b : a\in A, \ b \in B\}$. A. Sárközy has recently conjectured that the set $Q$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.

math.NT↗
arXiv2014

Subgroups Generated by Rational Functions in Finite Fields

For a large prime $p$, a rational function $ψ\in F_p(X)$ over the finite field $F_p$ of $p$ elements, and integers $u$ and $H\ge 1$, we obtain a lower bound on the number consecutive values $ψ(x)$, $x = u+1, \ldots, u+H$ that belong to a given multiplicative subgroup of $F_p^*$.

math.NT↗
arXiv2014

Circulant graphs and GCD and LCM of Subsets

Given two sets $A$ and $B$ of integers, we consider the problem of finding a set $S \subseteq A$ of the smallest possible cardinality such the greatest common divisor of the elements of $S \cup B$ equals that of those of $A \cup B$. The particular cases of $B = \emptyset$ and $\#B = 1$ are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Minimum Cover Problem.

cs.CC↗
arXiv2014

Double Character Sums over Subgroups and Intervals

We estimate double sums $$ S_χ(a, I, G) = \sum_{x \in I} \sum_{λ\in G} χ(x + aλ), \qquad 1\le a < p-1, $$ with a multiplicative character $χ$ modulo $p$ where $I= \{1,\ldots, H\}$ and $G$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_χ(a, I, G)$ can be derived from the Burgess bound if $H \ge p^{1/4+\varepsilon}$ and from some standard elementary arguments if $T \ge p^{1/2+\varepsilon}$, where $\varepsilon>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$ T_χ(a, G) = \sum_{λ, μ\in G} χ(a + λ+ μ), \qquad 1\le a < p-1, $$ and give an application to primitive roots modulo $p$ with $3$ non-zero binary digits.

math.NT↗
arXiv2014

On the Greatest Common Divisor of Shifted Sets

Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.

math.NT↗
arXiv2014

Polynomial Values in Small Subgroups of Finite Fields

For a large prime $p$, and a polynomial $f$ over a finite field $F_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of $F_p^*$ containing $H\ge 1$ consecutive values $f(x)$, $x = u+1, \ldots, u+H$, uniformly over $f\in F_p[X]$ and an $u \in F_p$.

math.NT↗
arXiv2013

Quadratic Non-residues in Short Intervals

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+ψ(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $ψ(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.

math.NT↗
arXiv2013

On Gauss sums and the evaluation of Stechkin's constant

For the Gauss sums which are defined by S_n(a,q) := \sum_{x (mod q)} e(ax^n/q), Stechkin (1975) conjectured that the quantity A := \sup_{n,q\ge 2} \max_{\gcd(a,q)=1} |S_n(a,q)|/q^(1-1/n) is finite. Shparlinski (1991) proved that A is finite, but in the absence of effective bounds on the sums S_n(a,q) the precise determination of A has remained intractable for many years. Using recent work of Cochrane and Pinner (2011) on Gauss sums with prime moduli, in this paper we show that with the constant given by A = |S_6(4787,4606056)|/4606056^(5/6) = 4.709236... one has the sharp inequality |S_n(a,q)| \le Aq^(1-1/n) for all n,q \ge 2 and all integers a with gcd(a,q)=1. One interesting aspect of our method is that we apply effective lower bounds for the center density in the sphere packing problem due to Cohn and Elkies (2003) to optimize the running time of our primary computational algorithm.

math.NT↗
arXiv2013

Covering sets for limited-magnitude errors

For a set $\cM=\{-μ,-μ+1,\ldots, λ\}\setminus\{0\}$ with non-negative integers $λ,μ<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(λ,μ;q)$-\emph{covering set} if $$ \cM \cS=\{ms \bmod q : m\in \cM,\ s\in \cS\}=\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(λ,μ;q)$-covering set $\cS$ which is of the size $q^{1 + o(1)}\max\{λ,μ\}^{-1/2}$ for almost all integers $q\ge 1$ and of optimal size $p\max\{λ,μ\}^{-1}$ if $q=p$ is prime. Furthermore, using a bound on the fourth moment of character sums of Cochrane and Shi we prove the bound $$ω_{λ,μ}(q)\le q^{1+o(1)}\max\{λ,μ\}^{-1/2},$$ for any integer $q\ge 1$, however the proof of this bound is not constructive.

cs.IT↗
arXiv2013

Exponential Sums over Points of Elliptic Curves

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.

math.NT↗
arXiv2013

On the Number of Eisenstein Polynomials of Bounded Height

We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree $d$ and of height at most $H$, as $H\to \infty$. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most $H$.

math.NT↗
arXiv2013

On Shifted Eisenstein Polynomials

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree $n$ polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.

math.NT↗
arXiv2013

On Solutions to Some Polynomial Congruences in Small Boxes

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side length below $p^{1/2}$, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

math.NT↗