Real radical initial ideals

TL;DR

This paper investigates how real radical initial ideals influence the geometry of real varieties and the stability of sums of squares representations, providing algebraic criteria and applications in real algebraic geometry.

Contribution

It establishes conditions under which initial ideals are real radical and links these to stability of quadratic modules and sums of squares representations.

Findings

If in_w(I) is real radical, then w lies in the logarithmic set of the real variety.

Provides algebraic conditions for w to be in the logarithmic limit set of a semialgebraic set.

When w has positive entries, the quadratic module associated with I is stable.

Abstract

We explore the consequences of an ideal I of real polynomials having a real radical initial ideal, both for the geometry of the real variety of I and as an application to sums of squares representations of polynomials. We show that if in_w(I) is real radical for a vector w in the tropical variety, then w is in the logarithmic set of the real variety. We also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set. If in addition the entries of w are positive, then the corresponding quadratic module is stable. In particular, if in_w(I) is real radical for some positive vector w then the set of sums of squares modulo I is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer.