Lower bound functions for polynomials

Lower Bound Functions for Polynomials

Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation

This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. The coefficients of this expansion over a given box yi...

A lower bound for the root separation of polynomials

An experimental study by Collins (JSC; 2001) suggested the conjecture that the minimum separation of real zeros of irreducible integer polynomials is about the square root of Mahler’s bound for general integer polynomials. We prove that a power of about two thirds of the Mahler bound is already a lower bound for the minimum root separation of all integer polynomials.

Bernstein's polynomials for convex functions and related results

In this paper we establish several polynomials similar to Bernstein's polynomials and several refinements of  Hermite-Hadamard inequality for convex functions.

A Lower Bound for Average Values of Dynamical Green’s Functions

We provide a Mahler/Elkies-style lower bound for the average values of dynamical Green’s functions on P over an arbitrary valued field, and give some dynamical and arithmetic applications.