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

We find a L\'evy-Khinchin formula for radial functions on free groups. As a corollary we obtain a linear bound on the growth of radial, conditionally negative definite functions on free groups of two or more generators.

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.

We find a lower bound for $\chi = 1/p+1/q+1/r$ limiting any solution in the hyperbolic case of the Generalized Fermat Equation $x^p + y^q = z^r$.

We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes…

We prove new upper bounds for the length of laws that hold for all groups of size at most $n$ -- improving on previous results of Bou-Rabee and Kassabov-Matucci. The methods make use of the classification of finite simple groups. Stronger…

We describe the fully residually F; or limit groups relative to F; (where F is a free group) that arise from systems of equations in two variables over F that have coefficients in F.

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal…

We show that the theory of the free group -- and more generally the theory of any torsion-free hyperbolic group -- is $n$-ample for any $n\geq 1$. We give also an explicit description of the imaginary algebraic closure in free groups.

Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the…

$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$. The paper deals with the general problem of computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$, $i=1,...,m$.…

Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us…

We derive upper and lower bounds on the determinant of an exponential matrix. They can be transformed into corresponding bounds for the determinant of a univariate Gaussian matrix.

We study finite subgroups of outer automorphisms of free products. We give upper bounds for the orders of these finite subgroups as well as bounds for the orders of individual torsion outer automorphisms under some (necessary) conditions…

A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by…

A lower bound for the Gaussian Q-function is presented in the form of a single exponential function with parametric order and weight. We prove the lower bound by introducing two functions, one related to the Q-function and the other…

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…

We obtain a bound on the number of solutions of $x^q=x$ in a finite noncommutative algebra over a field with $q$ elements. Furthermore, we completely characterize those rings for which this maximum number is attained.

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…