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The generalized Lax conjecture for rigidly convex sets

From papers

Let p∈R[x]p\in\mathbb{R}[\mathbf{x}] be a real-zero (RZ) polynomial with p(0)=1p(\mathbf{0})=1. For a polynomial ff, write rcs⁡(f)\operatorname{rcs}(f) for its rigidly convex set. Generalized Lax conjecture. There exist another RZ polynomial q∈R[x]q\in\mathbb{R}[\mathbf{x}] with rcs⁡(p)⊆rcs⁡(q)\operatorname{rcs}(p)\subseteq\operatorname{rcs}(q) and symmetric matrices A1,…,AnA_{1},\dots,A_{n} such that

qp=det⁡(I+x1A1+⋯+xnAn).qp=\det(I+x_{1}A_{1}+\cdots+x_{n}A_{n}).

This conjecture softens the requirement that pp itself have a determinantal representation while preserving its rigidly convex set through the inclusion rcs⁡(p)⊆rcs⁡(q)\operatorname{rcs}(p)\subseteq\operatorname{rcs}(q). The supplied source describes the result as a conjecture to be proved, but gives no resolution evidence; its status is therefore recorded as open.

Progress summary

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The generalized Lax conjecture for rigidly convex sets

    Let S⊂\mathdsRnS\subset\mathds R^n be a rigidly convex set, meaning that it is an algebraic interior whose defining polynomial is real zero with respect to an interior point. A spectrahedron is the solution set of a linear matrix inequality.

    Generalized Lax conjecture. Every rigidly convex set is a spectrahedron.

    Spectrahedra are known to be rigidly convex, so this conjecture asks whether the converse holds. Its status is not established in the supplied text.

    source: Renato G. Bettiol, Mario Kummer and Ricardo A. E. Mendes, “Two results on the Convex Algebraic Geometry of sets with continuous symmetries”, arXiv:2408.03231 (2025).

Sources & referencesView supporting material

Primary source

Alejandro González Nevado, “The generalized Lax conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms”, arXiv:2601.12267 (2026).

Additional references

2 papers in this index state this conjecture (2025–2026). The statement above is taken from the most recent of them; the others are arXiv:2507.03800.

convex-algebraic-geometryconvexitydeterminantal-representationsreal-algebraic-geometryspectrahedra

Solutions 0

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