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Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic Sets

Zihao Dai, Zijia Li, Zhi-Hong Yang, Lihong Zhi

TL;DR

This work investigates the algebraic boundaries $\partial_aK$ of convex semi-algebraic sets and their connection to dual varieties. It refines Sinn's iterated singular-locus approach by establishing a Whitney $(a)$ stratification that captures all irreducible components $Z\subseteq\mathrm{Ex}_a(K)$ whose dual $Z^*$ appears as an irreducible component of $\overline{\partial_aK^\mathrm{o}}$, and it provides a Teissier-based algorithm using conormal spaces and prime decomposition to compute these stratifications. The authors prove that any such $Z$ must be an irreducible component of some $F_i$ in a canonical Whitney $(a)$ filtration and offer a practical procedure to compute the stratification via conormal-space ideals and saturation/prime-decomposition steps. They illustrate the method with Xano and Teardrop, showing extreme points whose dual components are revealed by Whitney $(a)$ stratification but missed by iterated singular loci, thereby giving a computationally effective tool for identifying algebraic boundaries of dual convex bodies. Overall, the results advance the computational toolkit for convex algebraic geometry by linking duality, stratification theory, and Teissier-style criteria to locate algebraic boundaries.

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.

Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic Sets

TL;DR

This work investigates the algebraic boundaries of convex semi-algebraic sets and their connection to dual varieties. It refines Sinn's iterated singular-locus approach by establishing a Whitney stratification that captures all irreducible components whose dual appears as an irreducible component of , and it provides a Teissier-based algorithm using conormal spaces and prime decomposition to compute these stratifications. The authors prove that any such must be an irreducible component of some in a canonical Whitney filtration and offer a practical procedure to compute the stratification via conormal-space ideals and saturation/prime-decomposition steps. They illustrate the method with Xano and Teardrop, showing extreme points whose dual components are revealed by Whitney stratification but missed by iterated singular loci, thereby giving a computationally effective tool for identifying algebraic boundaries of dual convex bodies. Overall, the results advance the computational toolkit for convex algebraic geometry by linking duality, stratification theory, and Teissier-style criteria to locate algebraic boundaries.

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.
Paper Structure (10 sections, 10 theorems, 47 equations, 2 figures, 2 algorithms)

This paper contains 10 sections, 10 theorems, 47 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Duality of Varieties and Convex Sets
  3. Conormal Spaces, Dual Varieties and Duality of Convex Sets
  4. Boundary of Convex Semi-Algebraic Sets
  5. Stratification on Varieties
  6. Stratification of Algebraic Boundaries of Convex Semi-Algebraic Sets
  7. Computing Whitney (a) stratifications via conormal spaces
  8. Examples
  9. Xano
  10. Teardrop

Key Result

Theorem 1.2

Let $K\subseteq\mathbb{R}^n$ be a semi-algebraic convex body with $0\in\mathrm{int}(K)$. Let $Z\subseteq\mathrm{Ex}_a(K)$ be an irreducible subvariety with $Z\cap\mathrm{Ex}(K)$ dense in $Z$ such that $\overline{Z}^*$ is an irreducible component of $\overline{\partial_aK^\mathrm{o}}$. Then $Z$ is an where $S(X,Y)$ is the set of points in which the pair $(X,Y)$ does not satisfy Whitney's condition

Figures (2)

  • Figure 1: Example \ref{['ex:tear']}.
  • Figure 2: Shifted Xano defined by $f = (x^4+(z+1)^3-(y+2)(z+1)^2)(y-1)$, and its dual defined by $f^* = (2 v_1+v_2-1)(v_0^4+128 v_1^4+320 v_1^3 v_2+256 v_1^2 v_2^2+64 v_1 v_2^3-64 v_1^3-128 v_1^2 v_2-64 v_1 v_2^2)$.

Theorems & Definitions (31)

  • Example 1.1
  • Theorem 1.2: Theorem \ref{['thm:strata']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 21 more