On semidefinite-representable sets over valued fields

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.

💡 Research Summary

The paper investigates the analogue of polyhedral and spectrahedral geometry when the underlying data are taken from a valued field (K, val). After fixing a complete discrete valued field (often exemplified by the p‑adic numbers Qₚ) with valuation ring O_K and residue field κ, the authors define a K‑polyhedron as a set of points x∈Kⁿ satisfying a finite collection of linear valuation inequalities val(ℓ_i(x))≥0 together with equalities val(m_j(x))=+∞, where ℓ_i and m_j are affine linear forms. In matrix notation this becomes {x | A x+v ⪰0, B x+w=0}. The first major result, Theorem 1.1 (Direct Image), states that for any affine map f:Kⁿ→Kᵐ, the image f(P) of a K‑polyhedron P is again a K‑polyhedron. The proof proceeds by decomposing an arbitrary linear map into three elementary types: (i) automorphisms in GL_n(O_K), (ii) diagonal scaling maps, and (iii) coordinate projections that drop the last variable. For automorphisms and diagonal maps the preservation of valuation inequalities is straightforward. The projection case is the technical core: the authors compute a Smith Normal Form (SNF) of the submatrix of A corresponding to the retained coordinates, thereby isolating invariant factors π^{a_i}. Using the ultrametric property of the valuation, they rewrite the last coordinate in terms of the others and show that the resulting constraints are again of the required valuation‑inequality form. By chaining these three lemmas, the general case follows.

Having established the stability of K‑polyhedra under linear images, the paper turns to linear programming over (K, val). Section 4 presents Algorithm 1, which solves a K‑LP problem by first putting the constraint matrix into SNF. This reduces the system to a set of decoupled valuation constraints on transformed variables. The algorithm then determines the minimal valuation feasible for each variable and evaluates the linear objective accordingly. Correctness follows from the fact that SNF preserves the module structure of the constraint matrix and that the valuation is a non‑Archimedean norm satisfying the strong triangle inequality. The authors note that SNF can be computed in polynomial time, implying that K‑LP enjoys a comparable computational complexity to classical real LP, at least in the exact arithmetic model.

The second half of the paper introduces K‑spectrahedra, defined as the solution sets of linear matrix inequalities (LMIs) A₀+∑_{i=1}^n x_i A_i ⪰ 0 where the matrices A_i have entries in K and “⪰0” means that every eigenvalue has non‑negative valuation in an algebraic closure of K. This mirrors the real case, where positivity of eigenvalues characterizes semidefinite cones. The authors prove that K‑spectrahedra are convex, basic semialgebraic sets (with respect to the valuation) and that they contain K‑polyhedra as a special case (by choosing diagonal matrices). They then study the class of semidefinite‑representable (SDR) sets, i.e., linear projections of K‑spectrahedra, and ask whether every SDR set is itself a spectrahedron.

A striking negative result is given in Theorem 1.2: when the residue field κ is finite, any non‑trivial annulus {x∈K | α ≤ val(x) ≤ β} (with α<β) is SDR but cannot be a K‑spectrahedron. The proof exploits the finiteness of κ to show that any K‑spectrahedron’s defining LMI would force the valuation of eigenvalues to lie in a single coset of the value group, contradicting the width of the annulus. Consequently, the paper exhibits explicit one‑dimensional SDR sets that are not spectrahedral, extending the known separation between spectrahedral shadows and spectrahedra from the real to the valued setting.

Overall, the work establishes a robust framework for convex geometry and optimization over valued fields. It generalizes classical results—such as the closure of polyhedra under linear maps and the existence of exact LP algorithms—to the non‑Archimedean context by leveraging Smith Normal Forms and valuation theory. Moreover, it reveals new phenomena unique to valued fields, notably the existence of SDR sets that cannot be expressed as spectrahedra, thereby enriching the landscape of semidefinite‑representability. The authors suggest that these techniques could be extended to more general (possibly non‑discrete) valuations, to tropical geometry, and to applications in p‑adic optimization and number‑theoretic semidefinite programming.