The lax conjecture is true

By A. S. Lewis, Pablo A. Parrilo, M. Ramana. 2003

190 citations recorded by the source corpus.

In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5e6d2448a15ec1907d308db708f82775f459f6c9

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

POSITIVE POLYNOMIALS IN SCALAR AND MATRIX VARIABLES, THE SPECTRAL THEOREM AND OPTIMIZATION (2006, background): “but what about the converse? When g = 2 the real zero condition on p insures that it has a monic representation; this is the core of [HVprept]. What about higher dimensions? Lewis, Parrilo and Ramana [LPR05] showed that this g = 2 result (together with a counterexample they concocted) settles a 1958”
Multivariate Pólya–Schur classification problems in the Weyl algebra (2006, background): “The proof relies on the Lax conjecture that was recently settled in [38] by using the results of [30] in an essential way.”
Semidefinite Representation of the k-Ellipse (2007, methodology): “Our last topic in this section is the relationship of the k-ellipse to the celebrated work of Helton and Vinnikov [5] on LMI representations of planar convex sets, which led to the resolution of the Lax conjecture in [8].”
Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem (2013, methodology): “le polynomials. It is stated in the form we want by Borcea and Brand´en [BB10, Corollary 6.7], and is proved using an adaptation of a result of Helton and Vinnikov [HV07] by Lewis, Parrilo and Ramana [LPR05]. Lemma 5.5. If p(z 1,z 2) is a bivariate real stable polynomial of degree exactly d, then th”
LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future (2012, background): “1 then provides a solution the Lax conjecture concerning homogeneous hyperbolic polynomials in three variables, see [38], whereas Conjecture 3.”
Obstructions to determinantal representability (2010, background): “1 also settles a conjecture of Peter Lax which asserts that any hyperbolic degree d polynomial in three variables can be represented by a determinant, see [10].”
Polynomial convolutions and (nite) free probability (2021, methodology): “An extremely useful characterization of hyperbolic polynomials [15, 14]: Theorem 5.”
Polynomials with the half-plane property and matroid theory (2006, background): “The independent sets of matroids are precisely those delta-matroids that are hereditary and sets of bases of matroids are precisely the delta-matroids for which all the members of F have the same cardinality.”