Related papers: A polynomial bound on solutions of quadratic equat…

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by $2s$ for orientable equation and by $12s^4$ for non-orientable equation, where $s$ is the sum of the…

We prove that in a torsion-free hyperbolic group $\Gamma$, the length of the value of each variable in a minimal solution of a quadratic equation $Q=1$ is bounded by $N|Q|^3$ for an orientable equation, and by $N|Q|^{4}$ for a…

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…

We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm…

We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and…

We provide an algorithm which, for a given quadratic equation in the Grigorchuk group determines if it has a solution. As a corollary to our approach, we prove that the group has a finite commutator width.

Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold.…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We give upper and lower bounds for the number of solutions of the equation $p(z)\log|z|+q(z)=0$ with polynomials $p$ and $q$.

We give a bound for the number of real solutions to systems of n polynomials in n variables, where the monomials appearing in different polynomials are distinct. This bound is smaller than the fewnomial bound if this structure of the…

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.

We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial…

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy…

We give upper bounds on the order of the automorphism group of a simple graph

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,…

We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity…

Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…

We consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup…

We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions $f:\{1, ..., N\}\to\{1, ..., M\}$, its polynomial degree is the same for all…