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Igor E. Shparlinski

Publications and source records attributed to Igor E. Shparlinski.

At least 127 records · Page 7Linked to original sources
arXiv2018

On multiplicative dependence of values of rational functions and a generalisation of the Northcott theorem

In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a broad generalisation of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of initial points with two multiplicatively dependent elements in their orbits.

math.NT↗
arXiv2018

Disjointness of the Möbius Transformation and Möbius Function

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation $x \mapsto (ax + b)/(cx + d)$ over a finite field of $p$ elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the Möbius function.

math.NT↗
arXiv2018

The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves

We obtain new results concerning the Sato-Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.

math.NT↗
arXiv2018

Sparsity of curves and additive and multiplicative expansion of rational maps over finite fields

For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated compositions of several rational functions are not contained in any short intervals or small subgroups.

math.NT↗
arXiv2018

Identity Testing and Interpolation from High Powers of Polynomials of Large Degree over Finite Fields

We consider the problem of identity testing and recovering (that is, interpolating) of a "hidden" monic polynomials $f$, given an oracle access to $f(x)^e$ for $x\in\mathbb F_q$, where $\mathbb F_q$ is the finite field of $q$ elements and an extension fields access is not permitted. The naive interpolation algorithm needs $de+1$ queries, where $d =\max\{{\rm deg}\ f, {\rm deg }\ g\}$ and thus requires $ de<q$. For a prime $q = p$, we design an algorithm that is asymptotically better in certain cases, especially when $d$ is large. The algorithm is based on a result of independent interest in spirit of additive combinatorics. It gives an upper bound on the number of values of a rational function of large degree, evaluated on a short sequence of consecutive integers, that belong to a small subgroup of $\mathbb F_p^*$.

cs.CC↗
arXiv2018

On abelian multiplicatively dependent points on a curve in a torus

We show, under some natural conditions, that the set of abelian (and thus also cyclotomic) multiplicatively dependent points on an irreducible curve over a number field is a finite union of preimages of roots of unity by a certain finite set of primitive characters from $\Gm^n$ to $\Gm$ restricted to the curve, and a finite set. We also introduce the notion of primitive multiplicative dependence and obtain a finiteness result for primitively multiplicatively dependent points defined over a so-called Bogomolov extension of a number field.

math.NT↗
arXiv2018

Sums with the Mobius function twisted by characters with powerful moduli

In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums of the Mobius function twisted by characters of the same type, complementing and improving some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of $L$-functions with the same class of moduli.

math.NT↗
arXiv2017

On the exponential large sieve inequality for sparse sequences modulo primes

We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $λ^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.

math.NT↗
arXiv2017

Denominators of Bernoulli polynomials

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with at most one exception. We also show that $\mathfrak{P}_n$ is large, has many prime factors exceeding $\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\sqrt{n}$ grows asymptotically as $κ\sqrt{n}/\log n$ for some constant $κ$ with $κ=2$. Further, we compare the sizes of $\mathfrak{P}_n$ and $\mathfrak{P}_{n+1}$, leading to the somewhat surprising conclusion that although $\mathfrak{P}_n$ tends to infinity with $n$, the inequality $\mathfrak{P}_n>\mathfrak{P}_{n+1}$ is more frequent than its reverse.

math.NT↗
arXiv2017

On Functional Graphs of Quadratic Polynomials

We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These include: the number of connected functional graphs, the number of graphs having a maximal cycle, the number of cycles of fixed size, the number of components of fixed size, as well as the shape of trees extracted from functional graphs. We particularly focus on connected functional graphs, that is, the graphs which contain only one component (and thus only one cycle). Based on the results of our computations, we formulate several conjectures highlighting the similarities and differences between these functional graphs and random mappings.

math.NT↗
arXiv2017

Character sums with smooth numbers

We use the large sieve inequality for smooth numbers due to S. Drappeau, A. Granville and X. Shao (2017), together with some other arguments, to improve their bounds on the frequency of pairs $(q,χ)$ of moduli $q$ and primitive characters $χ$ modulo $q$, for which the corresponding character sums with smooth numbers are large.

math.NT↗
arXiv2017

Connected components of the graph generated by power maps in prime finite fields

Consider the power pseudorandom-number generator in a finite field ${\mathbb F}_q$. That is, for some integer $e\ge2$, one considers the sequence $u,u^e,u^{e^2},\dots$ in ${\mathbb F}_q$ for a given seed $u\in {\mathbb F}_q^\times$. This sequence is eventually periodic. One can consider the number of cycles that exist as the seed $u$ varies over ${\mathbb F}_q^\times$. This is the same as the number of cycles in the functional graph of the map $x\mapsto x^e$ in ${\mathbb F}_q^\times$. We prove some estimates for the maximal and average number of cycles in the case of prime finite fields.

math.NT↗
arXiv2017

On the error term of a lattice counting problem

We improve the error terms of some estimates related to counting lattices from recent work of L. Fukshansky, P. Guerzhoy and F. Luca (2017). This improvement is based on some analytic techniques, in particular on bounds of exponential sums coupled with the use of Vaaler polynomials.

math.NT↗
arXiv2017

On short products of primes in arithmetic progressions

We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $α, β\le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots, p_k \le m^α$ are primes and $s \le m^β$ is a product of at most $\ell$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k = \ell =1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).

math.NT↗
arXiv2017

On two functions arising in the study of the Euler and Carmichael quotients

We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results concerning the relations between these functions as well as their typical and extreme values.

math.NT↗
arXiv2017

Reductions Modulo Primes of Systems of Polynomial Equations and Algebraic Dynamical Systems

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.

math.NT↗