The Generalized Lax Conjecture: is every hyperbolicity cone a spectrahedron?

The fundamental open question raised in Kummer, Plaumann and Vinzant's "Hyperbolic polynomials, interlacers, and sums of squares" (Mathematical Programming 153, 2015) is: can every hyperbolic program be rewritten as a semidefinite program, i.e. is every hyperbolicity cone a spectrahedron? This is the Generalized Lax Conjecture (GLC) in its modern, cone-level sense. The paper realized every hyperbolicity cone as a linear slice of the cone of nonnegative polynomials, via Wronskians of interlacers, and built sums-of-squares relaxations that are projections of spectrahedra, but could not decide exact semidefinite representability. GLC remains open in full generality as of August 2026; a January 2026 preprint claiming a full proof is unverified and uncited, and is not treated as closure.

Background

A real homogeneous polynomial $f\in\mathbb{R}[x_1,\dots,x_n]$ is called hyperbolic with respect to a point $e\in\mathbb{R}^n$ with $f(e)\neq0$ if for every $x\in\mathbb{R}^n$ the univariate polynomial $t\mapsto f(x+te)$ has only real roots. The notion goes back to the theory of partial differential equations; Gårding proved that the associated hyperbolicity cone $C(f,e)$, defined as the closure of the connected component of $\{x : f(x)\neq0\}$ containing $e$, is a convex cone. The prototype is $f(x)=\det(x_1A_1+\dots+x_nA_n)$ for real symmetric matrices $A_i$ with $\sum_i e_iA_i$ positive definite: such an $f$ is hyperbolic with respect to $e$ and its cone is $C(f,e)=\{x : \sum_i x_iA_i\succeq0\}$, where $\succeq0$ denotes positive semidefiniteness. A convex cone of this form, given by a homogeneous linear matrix inequality (LMI), is called a spectrahedral cone; linear optimization over an affine slice of a spectrahedral cone is a semidefinite program (SDP), a central model of convex optimization with mature algorithms.

Conversely, Güler and Renegar proposed hyperbolic programming — linear optimization over affine slices of arbitrary hyperbolicity cones — as a class of convex programs that contains linear and semidefinite programming, and asked whether it is genuinely larger. In three variables it is not: Helton and Vinnikov proved the Lax conjecture, namely that every hyperbolic ternary form has, up to sign, a definite determinantal representation $f=\det(\sum_i x_iM_i)$ with real symmetric $M_i$ and $\sum_i e_iM_i\succ0$. In four or more variables such a representation need not exist for $f$ itself, and the question survives at two equivalent levels (the equivalence is standard, cf. Kummer's note on the specialized Vámos polynomial following Helton–Vinnikov): the cone level — is every hyperbolicity cone a spectrahedral cone? — and the multiple level — does every hyperbolic $f$ admit a hyperbolic multiple $q\cdot f$ (with $q$ hyperbolic and $C(f,e)\subseteq C(q,e)$) that has a definite determinantal representation? A positive answer is the Generalized Lax Conjecture (GLC); it says exactly that hyperbolic programming and semidefinite programming have the same expressive power at the level of feasible cones. The name is used here in the modern sense of Netzer–Sanyal and Scheiderer; some earlier references, the source paper included, attach the phrase "generalized Lax conjecture" to the stronger statement that powers of $f$ itself admit definite determinantal representations — the statement Brändén refuted — and that refuted powers version is not the conjecture meant here.

Kummer, Plaumann and Vinzant approached GLC through interlacing. A form $g$ of degree $d-1$ interlaces a hyperbolic form $f$ of degree $d$ with respect to $e$ if for every $x\in\mathbb{R}^n$ the roots of $g(x+te)$ interlace (alternate with) the roots of $f(x+te)$. The interlacers of $f$ form a convex cone $\mathrm{Int}(f,e)$, and Kummer, Plaumann and Vinzant expressed it through the Wronskian-type forms $\Delta_{e,a}f = D_ef\cdot D_af - f\cdot D_eD_af$, where $D_a$ is the directional derivative along $a$: for square-free $f$ one has $C(f,e)=\{a\in\mathbb{R}^n : \Delta_{e,a}f \geq 0 \text{ on } \mathbb{R}^n\}$, realizing the hyperbolicity cone as a linear slice of the cone of nonnegative polynomials. Strengthening nonnegativity to being a sum of squares (sos), i.e. $F=\sum_j q_j^2$, equivalently $F=v^\top Gv$ for some positive semidefinite Gram matrix $G$, yields the inner approximation $\{a : \Delta_{e,a}f \text{ is sos}\}$, a projection of a spectrahedron, and a hierarchy $C_N=\{a : (\sum_i x_i^2)^N\Delta_{e,a}f \text{ is sos}\}$ whose union is dense in $C(f,e)$ when the real zero set of $f$ is smooth, by Reznick's theorem on uniform denominators in Hilbert's seventeenth problem. The paper also notes a second relaxation $\{a : D_ef\cdot D_af \text{ is sos modulo the ideal } (f)\}$ and asks whether it always coincides with the first. Both relaxations are exact whenever some power of $f$ has a definite determinantal representation, but not in general: for the Vámos polynomial $h$, the bases-generating polynomial of the Vámos matroid (a multiaffine real-stable quartic in eight variables, shown stable by Wagner and Wei; stable means nonvanishing whenever every variable has strictly positive imaginary part), the paper proves that $\Delta_{78}h$ is neither sos nor sos modulo $(h)$, recovering Brändén's theorem that no power of $h$ has a definite determinantal representation — a refutation of the stronger algebraic conjecture of Helton and Vinnikov that powers always suffice, but not of GLC itself. Kummer later showed that the hyperbolicity cone of the specialized Vámos polynomial is nevertheless a spectrahedron, via an explicit $7\times7$ LMI with integer data, so the best-known candidate counterexample fails.

The state of the art around GLC is sharply two-sided. Positive: besides the three-variable case, spectrahedrality is known for hyperbolicity cones of elementary symmetric polynomials (Brändén), of multivariate matching polynomials (Amini), of the first derivative relaxation of the positive semidefinite cone (Saunderson), and of symmetric hyperbolic cubics (Blekherman, Lindberg, Shu). For cones with smooth boundary, Netzer and Sanyal proved every hyperbolicity cone is a spectrahedral shadow, i.e. a linear image of a spectrahedron; Scheiderer sharpened this in 2025 to second-order cone representability for Nash-smooth cones. Negative-adjacent: Brändén's Vámos example and the non-representable hyperbolic matroids of Amini and Brändén (Non-Pappus, Non-Desargues) refute algebraic strengthenings of GLC; Raghavendra, Ryder, Srivastava and Weitz proved that any semidefinite representation of general degree-$d$ hyperbolicity cones in $n$ variables must have dimension $(n/d)^{\Omega(d)}$, so representations cannot be small; and the non-shadow obstruction techniques of Bodirsky, Kummer and Thom apply to cones such as the copositive cone that are not hyperbolicity cones. Whether the unrestricted conjecture holds remains, in Scheiderer's 2025 formulation, wide open.

Problem Statement

Decide, in full generality and by a rigorous proof, the Generalized Lax Conjecture: for every real homogeneous polynomial $f$ that is hyperbolic with respect to some point $e\in\mathbb{R}^n$ (with arbitrary number of variables $n$, arbitrary degree, and no smoothness, symmetry, or irreducibility assumption), does the closed hyperbolicity cone $C(f,e)$ admit a representation as a spectrahedral cone, i.e. do there exist an integer $m$ and real symmetric $m\times m$ matrices $A_1,\dots,A_n$ with $\sum_i e_iA_i\succ0$ such that

$$C(f,e)\;=\;\{x\in\mathbb{R}^n : x_1A_1+\dots+x_nA_n\succeq0\}?$$

An affirmative answer must establish the representation for all hyperbolic polynomials; a negative answer must exhibit one explicit hyperbolic polynomial together with a proof that its hyperbolicity cone is not a spectrahedron of any size. The following formulations are known to be equivalent to the above: (i) every hyperbolic program can be exactly rewritten as a semidefinite program by a representation of the feasible cone that is uniform in the objective and the affine constraints; and (ii) for every $f$ hyperbolic with respect to $e$ there exists a hyperbolic polynomial $q$ with $C(f,e)\subseteq C(q,e)$ such that the product $q\cdot f$ has a definite symmetric determinantal representation. Note that the per-instance solvability of individual hyperbolic programs by some semidefinite program, which is known, does not answer this question; the required answer concerns the cone itself. No restriction on variables, degree, smoothness, or matroidal origin may be imposed on the claimed answer.

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