Lax conjecture

Lax formulated the conjecture in 1958 for symmetric determinantal representations of homogeneous hyperbolic polynomials in three variables. That original three-variable statement is now a theorem; the broader many-variable analogue is a separate, unsettled claim.

Known results

• The original three-variable statement was identified with the Helton-Vinnikov theorem, proved in 2007.
• Lewis, Parrilo, and Ramana published “The Lax conjecture is true” in 2005, with the proof available as arXiv:math/0304104.
• For more than three variables, the exact analogue fails; a quadratic counterexample is given for $n>3$.
• Brändén’s 2010 counterexamples show that the generalized determinantal-representation claim is false, while weaker power-representation questions remain distinct.

January 2026 generalized-conjecture proof claim

A January 2026 preprint claims that every real-zero polynomial acquires a monic symmetric determinantal representation after suitable determinantal multiplication, concluding that the generalized Lax conjecture is true. This conflicts with the cited 2010 counterexamples and has no independent verification or referee assessment in the retrieved material.

Current status (as of August 2026): The original three-variable Lax conjecture is settled, while the broader generalized conjecture remains disputed because its January 2026 proof claim is unverified and conflicts with earlier counterexamples.