Related papers: A polynomial bound on solutions of quadratic equat…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square…
We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables.…
We give upper bounds on the minimal degree of a model in $\mathbb{P}^2$ and the minimal bidegree of a model in $\mathbb{P}^1 \times \mathbb{P}^1$ of the curve defined by a given Laurent polynomial, in terms of the combinatorics of the…
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why…
Under reasonable assumptions, a group action on a module extends to the minimal free resolutions of the module. Explicit descriptions of these actions can lead to a better understanding of free resolutions by providing, for example,…
We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our…
We give the lower bound for the growth of the maximum value for a solution to the minimal surface equation with 0 boundary values over an unbounded simply connected domain.
We consider the quadratic optimization problem $\max_{x \in C}\ x^T Q x + q^T x$, where $C\subseteq\mathbb{R}^n$ is a box and $r := \mathrm{rank}(Q)$ is assumed to be $\mathcal{O}(1)$ (i.e., fixed). We show that this case can be solved in…
In this paper we establish three results on small-height zeros of quadratic polynomials over $\overline{\mathbb Q}$. For a single quadratic form in $N \geq 2$ variables on a subspace of $\overline{\mathbb Q}^N$, we prove an upper bound on…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
The asymptotic bound for a length-based attack on the Conjugacy Search Problem in relatively hyperbolic groups is cubic for hyperbolic elements and a "small" polynomial for parabolic elements, depending on the Conjugacy Search Problem for…
We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number
In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…
We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and…