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On semidefinite-representable sets over valued fields

Corentin Cornou, Simone Naldi, Tristan Vaccon

TL;DR

The paper generalizes polyhedra and semidefinite representations to complete discrete valued fields $(K,\mathrm{val})$, proving that affine images of $K$-polyhedra remain polyhedra and giving a Smith Normal Form–based algorithm for exact linear programming over $K$. It extends spectrahedra to valued fields by defining PSD via eigenvalues in an algebraic closure and shows that SDR sets can exist beyond spectrahedra, with annuli providing a concrete 1D counterexample when the residue field is infinite. In addition to the SDPs-generalization, the work highlights structural results such as the NSP-like integrality criterion for PSD matrices and the fact that polyannuli are SDR but not spectrahedra in certain cases, underscoring both the reach and limits of semidefinite representations over non-Archimedean fields.

Abstract

Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field $K$. For $K$-polyhedra and linear programming over $K$ we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral $K$-spectrahedra, as well as sets that are semidefinite-representable over $K$ but are not $K$-spectrahedra.

On semidefinite-representable sets over valued fields

TL;DR

The paper generalizes polyhedra and semidefinite representations to complete discrete valued fields , proving that affine images of -polyhedra remain polyhedra and giving a Smith Normal Form–based algorithm for exact linear programming over . It extends spectrahedra to valued fields by defining PSD via eigenvalues in an algebraic closure and shows that SDR sets can exist beyond spectrahedra, with annuli providing a concrete 1D counterexample when the residue field is infinite. In addition to the SDPs-generalization, the work highlights structural results such as the NSP-like integrality criterion for PSD matrices and the fact that polyannuli are SDR but not spectrahedra in certain cases, underscoring both the reach and limits of semidefinite representations over non-Archimedean fields.

Abstract

Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field . For -polyhedra and linear programming over we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral -spectrahedra, as well as sets that are semidefinite-representable over but are not -spectrahedra.
Paper Structure (10 sections, 23 theorems, 40 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 23 theorems, 40 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Preliminaries
  4. Polyhedra
  5. Projections and direct image of polyhedra
  6. Linear programming
  7. Spectrahedra
  8. Positive semidefinite matrices
  9. Definition of spectrahedra
  10. Polyannuli

Key Result

Theorem 1.1

Let $f : K^n \to K^m$ be an affine map and $\mathcal{P}$ a polyhedron in $K^n.$ Then $f(\mathcal{P})$ is a polyhedron in $K^m$.

Figures (1)

  • Figure 1: Strategy of proof of Theorem \ref{['theo:DI']}.

Theorems & Definitions (65)

  • Theorem 1.1: direct image (DI)
  • Theorem 1.2
  • Definition 2.1: Smith Normal Form
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3: DI${}_{\mathrm{aut}}$
  • proof
  • ...and 55 more