A sufficient condition for local nonnegativity

TL;DR

This paper establishes a sufficient condition for a real polynomial to be locally nonnegative at a point, based on the positivity of its Newton principal part's faces, aiding in local nonnegativity verification.

Contribution

It introduces a new criterion involving Newton's principal part for determining local nonnegativity of polynomials at a point.

Findings

If all $F$-faces of $f_N$ are strictly positive, then $f$ is locally nonnegative at the origin.

The criterion simplifies checking local nonnegativity using Newton's principal part.

Provides a practical condition for analyzing polynomial nonnegativity in real algebraic geometry.

Abstract

A real polynomial $f$ is called local nonnegative at a point $p$, if it is nonnegative in a neighbourhood of $p$. In this paper, a sufficient condition for determining this property is constructed. Newton's principal part of $f$ (denoted as $f_N$) plays a key role in this process. We proved that if every $F$-face, $(f_N)_F$, of $f_N$ is strictly positive over $(\mathbb{R}\setminus 0)^n$, then $f$ is local nonnegative at the origin $O$.