Related papers: Deciding lower-boundedness of polynomials

We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.

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…

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…

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…

A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the…

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…

The number of essentially different square polyominoes of order n and minimum perimeter p(n) is enumerated.

We establish a version of the Pommerenke-Levin-Yoccoz inequality for the modulus of a polynomial-like restriction of a global polynomial and give two applications. First it is shown that if the modulus of a polynomial-like restriction of an…

Define a subset of the complex plane to be a Rolle's domain if it contains (at least) one critical point of every complex polynomial P such that P(-1)=P(1). Define a Rolle's domain to be minimal if no proper subset is a Rolle's domain. In…

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller…

We establish the boundedness character of solutions of a system of rational difference equations with a variable coefficient

We present a new proof rule for verifying lower bounds on quantities of probabilistic programs. Our proof rule is not confined to almost-surely terminating programs -- as is the case for existing rules -- and can be used to establish…

To the best of our knowledge this paper is the first attempt to introduce and study polynomial interpolation of the polynomial data given on arbitrary varieties. In the first part of the paper we present results on the solvability of such…

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum…

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 $f(x)$, $x = u+1, \ldots, u+H$,…

A short note on bounds on distance to variety of a point in terms of the Taylor coefficients at the point.

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating…

We are interested in finding a nonlinear polynomial $P$ on $\mathbb{R}^n$ that solves the minimal surface equation. Even though no explicit solution is found in this article, we investigate constraints that a polynomial solution must obey.…

We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is…