Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic Sets

TL;DR

This paper refines the stratification of algebraic boundaries of convex semi-algebraic sets to Whitney (a) stratifications and introduces an algorithm for their computation, enhancing understanding of dual convex bodies.

Contribution

It advances the stratification theory of algebraic boundaries by establishing Whitney (a) stratifications and provides a practical algorithm for their computation.

Findings

Refined stratification of algebraic boundaries to Whitney (a) stratifications.

Developed an algorithm based on Teissier's criterion for computing these stratifications.

Improved understanding of the duality in convex semi-algebraic sets.

Abstract

Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn's work, we refine the stratification of iterated singular loci to a Whitney (a) stratification, which gives a list of candidates of varieties whose dual is an irreducible component of the algebraic boundary of the dual convex body. We also present an algorithm based on Teissier's criterion to compute Whitney (a) stratifications, which employs conormal spaces and prime decomposition.