Extending Theorems of Boros and Menzer

TL;DR

This paper extends theorems by Boros and Menzer on generalized polynomials and monomials, focusing on their zero-product and non-negativity conditions, using measure-category methods.

Contribution

It generalizes previous results by incorporating mixed measure-category largeness, broadening the scope of the original theorems.

Findings

Extended theorems to mixed measure-category largeness

Applied similar methods to generalized polynomials and monomials

Provided new conditions for zero-product and inequality cases

Abstract

We extend results of Boros and Menzer on the alternative equation $f(x)f(y)=0$ for generalized polynomials $f$, and their theorems on the conditional inequality $f(x)f(y)\ge 0$ for generalized monomials $f$ of even degree. We use similar methods and ideas. We replace the largeness, of the respective Borel plane set $D$, in the measure or in the Baire category sense, by its largeness in the mixed measure-category sense.