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arxiv: 2602.09702 · v2 · submitted 2026-02-10 · 🧮 math.AG · cs.SC· math.OC

On semidefinite-representable sets over valued fields

Corentin Cornou , Simone Naldi , Tristan Vaccon This is my paper

Pith reviewed 2026-05-16 05:22 UTC · model grok-4.3

classification 🧮 math.AG cs.SCmath.OC
keywords valued fieldsspectrahedrasemidefinite programmingpolyhedraSmith normal formlinear programmingnon-archimedean geometry
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The pith

Semidefinite-representable sets over valued fields K retain their key separation from K-spectrahedra and include non-polyhedral examples.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the study of polyhedra and spectrahedra from the reals to an arbitrary valued field K. It first gives an algorithm for solving linear programming over K that relies on computing Smith normal forms. It then shows that the basic structural facts about semidefinite-representable sets survive the change of scalars: there exist sets that are images of K-spectrahedra under linear maps yet are not themselves K-spectrahedra, and there exist K-spectrahedra that fail to be polyhedral. These facts establish that the hierarchy of representable convex sets remains strict in the valued setting.

Core claim

Semidefinite-representable sets over a valued field K are defined exactly as over the reals, by taking linear images of sets of positive-semidefinite matrices with entries in K. The paper proves that this family properly contains the K-spectrahedra themselves and that individual K-spectrahedra need not be polyhedral. An auxiliary algorithm based on Smith normal forms solves the associated linear programs over K.

What carries the argument

K-spectrahedron: the set of points in K^n such that a symmetric matrix whose entries are affine functions of the point is positive semidefinite with respect to the valuation on K.

If this is right

  • Linear programming over valued fields admits a deterministic algorithm via Smith normal forms.
  • The distinction between spectrahedra and their linear images persists, so the SDP hierarchy remains strict over K.
  • Convex optimization problems over K inherit the same duality and closure properties that hold over the reals.
  • Non-polyhedral spectrahedra exist in every valued field, providing strictly richer feasible sets than polyhedra alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same constructions may supply feasible sets for optimization problems arising in p-adic or tropical geometry.
  • Hybrid algorithms could combine the Smith-normal-form method for the linear part with semidefinite relaxations over the residue field.
  • The separation between K-spectrahedra and semidefinite-representable sets suggests that lift-and-project methods will require strictly more variables even in the valued setting.

Load-bearing premise

The usual definition of positive semidefiniteness and the notion of semidefinite representability extend directly to any valued field K without extra conditions on the valuation or the field characteristic.

What would settle it

An explicit K-spectrahedron over the p-adics that is polyhedral, or a concrete semidefinite-representable set over K that cannot be expressed as the projection of any K-spectrahedron.

Figures

Figures reproduced from arXiv: 2602.09702 by Corentin Cornou, Simone Naldi, Tristan Vaccon.

Figure 1
Figure 1. Figure 1: Strategy of proof of Theorem 1.1. We begin with the case of linear maps in GL𝑛 (O𝐾 ). We often fix a linear basis 𝜉1, . . . , 𝜉𝑛 of 𝐾 𝑛 which we call canonical. Lemma 3.3 (DIaut). Let 𝑓 ∈ GL𝑛 (O𝐾 ) and P a polyhedron in 𝐾 𝑛 . Then 𝑓 (P) is a polyhedron. If in addition, P is defined only by inequalities, then so is 𝑓 (P). Proof. Assume P is defined in matrix form as in (2), and let 𝑈 be the matrix of 𝑓 in t… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies analogues of polyhedra and spectrahedra over a valued field K. It gives an algorithm for K-polyhedra and linear programming over K based on Smith normal forms, proves that fundamental properties of semidefinite-representable sets extend to the valued setting, and exhibits examples of non-polyhedral K-spectrahedra together with sets that are semidefinite-representable over K but are not K-spectrahedra.

Significance. If the definitions and proofs are correct, the work extends the theory of spectrahedra and semidefinite representability from the reals to arbitrary valued fields, supplying both algorithmic tools for the polyhedral case and concrete examples that separate the notions of K-spectrahedra and K-SDR sets. Such results would be of interest to researchers working at the interface of real algebraic geometry, non-archimedean geometry, and optimization.

major comments (2)
  1. [Section 2 (definitions)] The central claims rest on a well-defined notion of positive semidefiniteness over an arbitrary valued field K. Standard spectrahedra over R rely on the canonical ordering; for general K (e.g., Q_p or Laurent series) no such ordering is given. The manuscript must explicitly state the definition of the PSD cone (for instance via valuation conditions on principal minors or an auxiliary ordering) and verify that it yields a proper convex cone compatible with the subsequent proofs.
  2. [Section 3 (algorithm)] The Smith-normal-form algorithm for K-polyhedra is stated to work for general valued fields. Smith normal form over the valuation ring requires the ring to be a PID, which holds only for discrete valuations. For non-discrete valuations the algorithm as described may fail; the paper should either restrict to discrete valuations or supply a replacement procedure.
minor comments (2)
  1. [Abstract] The abstract refers to 'examples' without citing the relevant theorems or sections; adding forward references would improve readability.
  2. [Throughout] Notation for the valued field K and its valuation ring should be introduced once and used consistently; occasional switches between K and its completion are confusing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the work's significance at the interface of real algebraic geometry, non-archimedean geometry, and optimization. We address each major comment below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

  1. Referee: [Section 2 (definitions)] The central claims rest on a well-defined notion of positive semidefiniteness over an arbitrary valued field K. Standard spectrahedra over R rely on the canonical ordering; for general K (e.g., Q_p or Laurent series) no such ordering is given. The manuscript must explicitly state the definition of the PSD cone (for instance via valuation conditions on principal minors or an auxiliary ordering) and verify that it yields a proper convex cone compatible with the subsequent proofs.

    Authors: We agree that the definition of positive semidefiniteness requires more explicit treatment for arbitrary valued fields. In the manuscript, the PSD cone is implicitly defined via the condition that a symmetric matrix is positive semidefinite precisely when the valuation of each of its principal minors is non-negative. This induces the required convex cone structure without relying on an ordering of K. To address the comment, we will add a dedicated paragraph in Section 2 that states this definition formally, proves that the resulting set is a proper convex cone, and verifies compatibility with the subsequent results on spectrahedra and semidefinite representability. The revision will also include a brief comparison with the real case. revision: yes

  2. Referee: [Section 3 (algorithm)] The Smith-normal-form algorithm for K-polyhedra is stated to work for general valued fields. Smith normal form over the valuation ring requires the ring to be a PID, which holds only for discrete valuations. For non-discrete valuations the algorithm as described may fail; the paper should either restrict to discrete valuations or supply a replacement procedure.

    Authors: This observation is correct: the Smith normal form computation over the valuation ring presupposes that the ring is a PID, which holds if and only if the valuation is discrete. The manuscript's algorithm and its complexity claims are therefore valid only under this hypothesis. In the revision we will explicitly restrict the statements of the algorithm, the linear-programming procedure, and the associated theorems in Section 3 to discrete valuations. We will add a remark clarifying that the non-discrete case lies outside the scope of the present work and would require different techniques based directly on the value group. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper defines K-polyhedra and K-spectrahedra over valued fields K by direct extension of the standard notions, then proves extension of fundamental properties and exhibits explicit examples using Smith normal forms and algebraic constructions. No equations or central claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the results rest on independent algebraic arguments applied to the new definitions. The work is self-contained against external benchmarks and does not invoke author-specific uniqueness theorems or ansatzes smuggled via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the classical definitions of polyhedra, spectrahedra, and semidefinite representability extend directly to valued fields; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Standard definitions of spectrahedra and semidefinite representability over the reals extend verbatim to an arbitrary valued field K.
    Invoked implicitly when the paper states that fundamental properties extend and when examples are exhibited.

pith-pipeline@v0.9.0 · 5437 in / 1292 out tokens · 42887 ms · 2026-05-16T05:22:11.333349+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

  1. [1]

    Allamigeon, S

    X. Allamigeon, S. Gaubert, and M. Skomra. 2016. Solving generic nonarchimedean semidefinite programs using stochastic game algorithms. InProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation. 31– 38

    work page 2016

  2. [2]

    Allamigeon, S

    X. Allamigeon, S. Gaubert, and M. Skomra. 2019. The tropical analogue of the Helton–Nie conjecture is true.Journal of Symbolic Computation91 (2019), 129–148

    work page 2019

  3. [3]

    Allamigeon, S

    X. Allamigeon, S. Gaubert, and M. Skomra. 2020. Tropical spectrahedra.Discrete & Computational Geometry63, 3 (April 2020), 507–548. doi:10.1007/s00454-020- 00176-1 arXiv:1610.06746 [math]

    work page doi:10.1007/s00454-020- 2020

  4. [4]

    X. Caruso. 2017. Computations with p-adic numbers. doi:10.48550/arXiv.1701. 06794 arXiv:1701.06794 [cs, math]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1701 2017

  5. [5]

    Develin and B

    M. Develin and B. Sturmfels. 2004. Tropical convexity.Documenta Mathematica 9 (2004), 1–27

    work page 2004

  6. [6]

    A. J. Engler and A. Prestel. 2005.Valued fields. Springer Science & Business Media

    work page 2005

  7. [7]

    J. W. Helton and J. Nie. 2009. Sufficient and necessary conditions for semidefinite representability of convex hulls and sets.SIAM J. Optim.20, 2 (2009), 759–791

    work page 2009

  8. [8]

    Henrion, M

    D. Henrion, M. Korda, and J-B. Lasserre. 2020.The Moment-sos Hierarchy. Vol. 4. World Scientific

    work page 2020

  9. [9]

    Henrion, S

    D. Henrion, S. Naldi, and M. Safey El Din. 2016. Exact algorithms for linear matrix inequalities.SIAM Journal on Optimization26, 4 (2016), 2512–2539

    work page 2016

  10. [10]

    S. Naldi. 2018. Solving rank-constrained semidefinite programs in exact arith- metic.Journal of Symbolic Computation85 (2018), 206–223. doi:10.1016/j.jsc.2017. 07.009

    work page doi:10.1016/j.jsc.2017 2018

  11. [11]

    Naldi, M

    S. Naldi, M. Safey El Din, A. Taylor, and W. Wang. 2025. Solving generic parametric linear matrix inequalities. InProceedings of the 2025 International Symposium on Symbolic and Algebraic Computation. 267–276

    work page 2025

  12. [12]

    Nemirovski

    A. Nemirovski. 2007. Advances in convex optimization: conic programming. In Proceedings of the International Congress of Mathematicians Madrid, August 22–30,

    work page 2007

  13. [13]

    European Mathematical Society-EMS-Publishing House GmbH, 413–444

    work page

  14. [14]

    Netzer and D

    T. Netzer and D. Plaumann. 2023.Geometry of Linear Matrix Inequalities. Springer

    work page 2023

  15. [15]

    Porkolab and L

    L. Porkolab and L. Khachiyan. 1997. On the complexity of semidefinite programs. Journal of Global Optimization10, 4 (1997), 351–365

    work page 1997

  16. [16]

    J. Renegar. 1992. On the computational complexity and geometry of the first- order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals. Journal of symbolic computation13, 3 (1992), 255–299

    work page 1992

  17. [17]

    J. Rüth. 2015. Models of curves and valuations. PhD Dissertation, Universität Ulm. doi:10.18725/OPARU-3275

    work page doi:10.18725/oparu-3275 2015

  18. [18]

    Scheiderer

    C. Scheiderer. 2018. Spectrahedral shadows.SIAM Journal on Applied Algebra and Geometry2, 1 (2018), 26–44

    work page 2018

  19. [19]

    J-P. Serre. 1979.Local fields. Graduate Texts in Mathematics, Vol. 67. Springer- Verlag, New York-Berlin. viii+241 pages. Translated from the French by Marvin Jay Greenberg

    work page 1979