The Projected Lax Conjecture: Is Every Hyperbolicity Cone a Spectrahedral Shadow?

A homogeneous real polynomial is hyperbolic with respect to a direction if its restriction to every line through that direction has only real roots; by a theorem of Gårding the corresponding component of the nonvanishing set is a convex cone, the hyperbolicity cone. Spectrahedral cones — feasible cones of linear matrix inequalities, i.e. of semidefinite programs — are exactly the hyperbolicity cones of determinant polynomials, and Netzer and Sanyal (2012) proved that every hyperbolicity cone whose boundary is smooth is a spectrahedral shadow, that is, the linear projection of a spectrahedral cone (the feasible set of a lifted linear matrix inequality). Their method breaks down at boundary singularities — points of multiplicity greater than one, line-like vanishing of factors, self-intersections — and they posed the Projected Lax Conjecture: every hyperbolicity cone, singular boundary included, is a spectrahedral shadow. As of August 2026 the singular-boundary case remains open: no accepted proof or refutation exists, and a January 2026 preprint claiming a proof of the generalized Lax conjecture is unverified and uncited.

Background

A homogeneous real polynomial $h(x_1,\dots,x_n)$ is called hyperbolic with respect to a vector $e\in\mathbb{R}^n$ if $h(e)\neq 0$ and, for every $x\in\mathbb{R}^n$, the univariate polynomial $t\mapsto h(te-x)$ has only real zeros. Gårding (1959) proved that the connected component of the set $\{x : h(x)\neq 0\}$ that contains $e$ is an open convex cone; its closure $\Lambda_e(h)$ is the hyperbolicity cone of $h$. The model example is the determinant: $\det$ is hyperbolic with respect to the identity matrix $I$, and $\Lambda_I(\det)$ is the cone of positive semidefinite (PSD) symmetric matrices. Hyperbolic polynomials originated in the theory of partial differential equations (Petrovskii, Gårding) and now play a central role in convex algebraic geometry and optimization, largely because Güler (1997) showed that $-\log|h|$ is a self-concordant barrier — a smoothness-and-convexity property that enables polynomial-time interior-point methods — for $\Lambda_e(h)$, so that linear optimization over any hyperbolicity cone, called a hyperbolic program, is tractable.

A spectrahedral cone is a cone of the form $\{x\in\mathbb{R}^n : A(x)=\sum_{i=1}^n x_iA_i\succeq 0\}$, where the $A_i$ are real symmetric matrices of a common size, $\succeq 0$ denotes positive semidefiniteness, and $A(e)\succ 0$ for some $e$; these are exactly the feasible cones of linear matrix inequalities (LMIs) and hence of semidefinite programs. If $A(e)\succ 0$ then $\det A(x)$ is hyperbolic with $\Lambda_e(\det A)=\{x : A(x)\succeq 0\}$, so every spectrahedral cone is a hyperbolicity cone. Whether the converse holds in any form is the central question of this circle of ideas. Lax (1958) conjectured a determinantal representation for homogeneous polynomials in three variables hyperbolic in all directions; in the equivalent real-zero formulation of Lewis, Parrilo and Ramana (2005) — where an inhomogeneous polynomial $p$ with $p(0)\neq 0$ is a real-zero polynomial if $t\mapsto p(tx)$ has only real zeros for every $x$, and the associated rigidly convex set $C(p)$, the component of $\{p\neq 0\}$ containing the origin (convex by the same theorem of Gårding), plays the role of the dehomogenized hyperbolicity cone — the conjecture was confirmed through the representation theorem of Helton and Vinnikov (2007): every hyperbolicity cone in three variables is spectrahedral. For $n\geq 4$ variables a dimension count already shows that most hyperbolic polynomials are not of the form $\det(\sum_i x_iA_i)$, and Brändén (2011) proved a far stronger obstruction: using the theory of matroids with the half-plane property (matroids whose basis-generating polynomial is real stable, i.e. has no zeros when all variables lie in the open upper half-plane), he constructed a real-zero polynomial — the specialized Vámos polynomial, derived from the Vámos matroid, a standard rank-4 counterexample configuration on eight elements — none of whose powers admits a definite determinantal representation, refuting the conjecture of Helton and Vinnikov that every real-zero polynomial has a symmetric determinantal representation. The conjectures therefore had to be weakened.

Two weakenings survive. The Generalized Lax Conjecture (GLC) asks that every hyperbolicity cone be spectrahedral; equivalently, for every $h$ hyperbolic with respect to $e$ there should exist a hyperbolic $q$ with interior inclusion $\Lambda_e^{++}(h)\subseteq\Lambda_e^{++}(q)$ (where $\Lambda^{++}$ denotes the interior of the cone) and a definite determinantal representation $q\cdot h=\det(\sum_i x_iA_i)$ with $\sum_i e_iA_i\succ 0$ (using the standard identity $\Lambda_e(qh)=\Lambda_e(q)\cap\Lambda_e(h)$). The Projected Lax Conjecture (PLC), posed by Netzer and Sanyal (2012), asks only that every hyperbolicity cone be a spectrahedral shadow — the linear projection of a spectrahedral cone, i.e. a set of the form $\{x : \exists y,\ \sum_i x_iA_i+\sum_j y_jB_j\succeq 0\}$, also called an sdp-representable or lifted-LMI set; optimization over such a set is still exactly a semidefinite program. Netzer and Sanyal proved the PLC under a smoothness hypothesis: if every nonzero boundary point of $\Lambda_e(h)$ is a smooth point of $h$ (the gradient of $h$ does not vanish there), which holds in particular for strictly hyperbolic polynomials — those whose univariate restrictions $h(te-x)$ have only simple roots, a dense open condition by a classical perturbation argument of Nuij (1968) — then $\Lambda_e(h)$ is a spectrahedral shadow. Their proof dehomogenizes the cone to a compact rigidly convex set described by real-zero polynomials, verifies strict quasi-concavity of the describing polynomials near smooth boundary points so that the semidefinite-representability machinery of Helton and Nie applies, and patches the local liftings into a global one. They emphasize that these tools break down in the presence of more severe boundary singularities, such as self-intersections, points where $h$ vanishes with multiplicity greater than one, or points where an irreducible factor of $h$ vanishes along an entire line; and that Nuij-type perturbation only approximates a general hyperbolicity cone by smooth ones, yielding approximating spectrahedral shadows but no exact representation. They close the paper by conjecturing the PLC in precisely the form stated here.

The PLC sits at the current boundary of knowledge about semidefinite representability: Scheiderer (2016) disproved the Helton–Nie conjecture by exhibiting compact convex semialgebraic sets that are not spectrahedral shadows, but all of his obstruction criteria fail to apply to hyperbolicity cones, which remain the prominent candidate class for universal shadow representability. On the positive side, Amini (2016) established the full GLC for the hyperbolicity cones of multivariate matching polynomials, and Scheiderer (2025) strengthened the Netzer–Sanyal theorem by proving that every hyperbolicity cone with Nash-smooth boundary — a semialgebraic smoothness condition slightly weaker than smoothness — is even second-order cone representable, meaning a lifted LMI whose blocks have size at most $2\times 2$ (the second-order, or Lorentz, cone being the hyperbolicity cone of a quadratic form). None of these results touches cones with genuinely singular boundary, and Scheiderer's paper states that even the weaker shadow question is completely open in general.

Problem Statement

The Projected Lax Conjecture of Netzer and Sanyal: is every hyperbolicity cone a spectrahedral shadow? Equivalently, for every real homogeneous polynomial $h$ that is hyperbolic with respect to a direction $e\in\mathbb{R}^n$, do there exist an integer $m$ and real symmetric matrices $A_1,\dots,A_n,B_1,\dots,B_m$ of a common size — the lifted spectrahedral cone being strictly feasible in the sense of the Background, i.e. $\sum_{i=1}^n x_iA_i+\sum_{j=1}^m y_jB_j\succ 0$ for some $(x,y)\in\mathbb{R}^{n+m}$ — such that

$$\Lambda_e(h)\;=\;\left\{\, x\in\mathbb{R}^n \;:\; \exists\, y\in\mathbb{R}^m,\ \sum_{i=1}^n x_iA_i+\sum_{j=1}^m y_jB_j\succeq 0 \,\right\}?$$

The crux is the removal of the smoothness hypothesis: the statement is known when every nonzero boundary point of $\Lambda_e(h)$ is a smooth point of $h$ (Netzer–Sanyal, strengthened by Scheiderer to Nash-smooth boundary, with blocks of size at most 2), and it is open precisely for hyperbolicity cones whose boundary contains singular points — points at which $h$ vanishes with multiplicity greater than one, points at which an irreducible factor of $h$ vanishes along an entire line, and self-intersections of the boundary. The conjecture is a bare existence statement, posed by Netzer and Sanyal in exactly this form: an affirmative answer establishes, for every hyperbolicity cone, the existence of matrices realizing the displayed lifted linear matrix inequality, with no requirement that they arise from a general constructive procedure or an efficiently checkable certificate; a negative answer must exhibit a hyperbolic polynomial whose hyperbolicity cone provably admits no lifted LMI representation of any size and any number of lifting variables. The stronger Generalized Lax Conjecture — the same question with no lifting variables, i.e. that every hyperbolicity cone is a spectrahedral cone — is likewise open; since every spectrahedral cone is a spectrahedral shadow, an affirmative answer to it settles the Projected Lax Conjecture a fortiori, whereas a negative answer to it — a hyperbolicity cone that is not spectrahedral — would leave the Projected Lax Conjecture open.

References

1. Tim Netzer and Raman Sanyal, 'Smooth hyperbolicity cones are spectrahedral shadows', Mathematical Programming (Series B) 153 (2015), no. 1, 213–221. DOI 10.1007/s10107-014-0744-6; arXiv:1208.0441. https://arxiv.org/abs/1208.0441
2. P. D. Lax, 'Differential equations, difference equations and matrix theory', Communications on Pure and Applied Mathematics 11 (1958), 175–194. DOI 10.1002/cpa.3160110203. https://doi.org/10.1002/cpa.3160110203
3. A. S. Lewis, P. A. Parrilo, and M. V. Ramana, 'The Lax conjecture is true', Proceedings of the American Mathematical Society 133 (2005), no. 9, 2495–2499. DOI 10.1090/S0002-9939-05-07752-X. https://doi.org/10.1090/S0002-9939-05-07752-X
4. J. William Helton and Victor Vinnikov, 'Linear matrix inequality representation of sets', Communications on Pure and Applied Mathematics 60 (2007), no. 5, 654–674. DOI 10.1002/cpa.20155. https://doi.org/10.1002/cpa.20155
5. Lars Gårding, 'An inequality for hyperbolic polynomials', Journal of Mathematics and Mechanics 8 (1959), 957–965. DOI 10.1512/iumj.1959.8.58061. https://doi.org/10.1512/iumj.1959.8.58061
6. Osman Güler, 'Hyperbolic polynomials and interior point methods for convex programming', Mathematics of Operations Research 22 (1997), no. 2, 350–377. DOI 10.1287/moor.22.2.350. https://doi.org/10.1287/moor.22.2.350
7. Petter Brändén, 'Obstructions to determinantal representability', Advances in Mathematics 226 (2011), no. 2, 1202–1212. DOI 10.1016/j.aim.2010.08.003; arXiv:1004.1382. https://arxiv.org/abs/1004.1382
8. Mario Kummer, 'A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial', Acta Applicandae Mathematicae 144 (2016), 11–15. DOI 10.1007/s10440-015-0036-z; arXiv:1306.4483. https://doi.org/10.1007/s10440-015-0036-z
9. Sam Burton, Cynthia Vinzant, and Yewon Youm, 'A real stable extension of the Vamos matroid polynomial', arXiv:1411.2038 (2014). https://arxiv.org/abs/1411.2038
10. Claus Scheiderer, 'Spectrahedral Shadows', SIAM Journal on Applied Algebra and Geometry 2 (2018), no. 1, 26–44. DOI 10.1137/17M1118981; arXiv:1612.07048. https://doi.org/10.1137/17M1118981
11. Nima Amini, 'Spectrahedrality of hyperbolicity cones of multivariate matching polynomials', Journal of Algebraic Combinatorics 50 (2019), no. 2, 165–190. DOI 10.1007/s10801-018-0848-9; arXiv:1611.06104. https://arxiv.org/abs/1611.06104
12. J. William Helton and Jiawang Nie, 'Semidefinite representation of convex sets', Mathematical Programming (Series A) 122 (2010), no. 1, 21–64. DOI 10.1007/s10107-008-0240-y. https://doi.org/10.1007/s10107-008-0240-y
13. Wim Nuij, 'A note on hyperbolic polynomials', Mathematica Scandinavica 23 (1968), 69–72. DOI 10.7146/math.scand.a-10898. https://doi.org/10.7146/math.scand.a-10898
14. Prasad Raghavendra, Nick Ryder, Nikhil Srivastava, and Benjamin Weitz, 'Exponential Lower Bounds on Spectrahedral Representations of Hyperbolicity Cones', Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), 2322–2332. DOI 10.1137/1.9781611975482.141. https://doi.org/10.1137/1.9781611975482.141
15. Rafael Oliveira, 'Conditional lower bounds on the spectrahedral representation of explicit hyperbolicity cones', Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation (ISSAC 2020), 396–401. DOI 10.1145/3373207.3404010. https://doi.org/10.1145/3373207.3404010
16. Markus Schweighofer, 'Spectrahedral relaxations of hyperbolicity cones', arXiv:1907.13611 (2019). https://arxiv.org/abs/1907.13611
17. David Sawall and Markus Schweighofer, 'Amalgamation of real zero polynomials', Indagationes Mathematicae 35 (2024), no. 1, 37–59. DOI 10.1016/j.indag.2023.08.002; arXiv:2305.07403. https://doi.org/10.1016/j.indag.2023.08.002
18. Claus Scheiderer, 'Smooth hyperbolicity cones are second-order cone representable', arXiv:2509.17121 (2025). https://arxiv.org/abs/2509.17121
19. 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). https://arxiv.org/abs/2601.12267