Related papers: Polynomial Equations in Subgroups and Applications

In this paper, we study positive integer solutions to a generalized form of the Markov equation, given as $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. This equation extends the classical Markov equation $x^2 + y^2 + z^2 = 3xyz$. We…

We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics,…

Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form…

We improve recent results of D. Gomez and A. Winterhof (2010) and of A. Ostafe and I. E. Shparlinski (2011) on the Waring problem with Dickson polynomials in the case of prime finite fields. Our approach is based on recent bounds of…

The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing…

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.

This is an expository account of the proof of the theorem of Bourgain, Glibichuk and Konyagin which provides non-trivial bounds for exponential sums over very small multiplicative subgroups of prime finite fields.

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric…

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent…

Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$, and let $\mathbb{Z}_n^{\ast}$ be the group of its units. 90 years ago, Brauer obtained a formula for the number of representations of $c\in \mathbb{Z}_n$ as the sum of $k$ units.…

We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an…

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a…

We consider the Markoff-Rosenberger equation $$ax^2+by^2+cz^2=dxyz$$ with $(x,y,z)=(U_i,U_j,U_k),$ where $U_i$ denotes the $i$-th generalized Lucas number of first/second kind. We provide upper bound for the minimum of the indices and we…

In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a…

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 $f(x)$, $x = u+1, \ldots, u+H$,…

We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements…

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the…