The Lax conjecture is true. (English) Zbl 1073.90029
A polynomial \(p\in\mathbb{H}^n(d)\), the set of homogeneous polynomials on \(\mathbb{R}^n\) of degree \(d\), is called hyperbolic with respect to \(e\in\mathbb{R}^n\) if \(p(e)\neq 0\) and for every \(w\in\mathbb{R}^n\) the univariate polynomial \(\widetilde p(t):= p(w-te)\) has all roots real. The corresponding hyperbolicity cone is the open convex cone \(\{w\in \mathbb{R}^n: \widetilde p(t)= 0\Rightarrow t> 0\}\). The set \(\{w\in\mathbb{R}^n: \sum w_jG_j\in \mathbb{S}^{d+}\}\), where the \(G_j\in\mathbb{S}^d\), the space of \(d\times d\) real symmetric matrices, and \(\mathbb{S}^{d+}\) is the positive definite cone, is called a semidefinite slice.
The authors prove that a nonempty semidefinite slice is a hyperbolicity cone by showing that if the semidefinite slice contains the vector \(\widetilde w\), then the polynomial \(p(w):= \text{det\,}\Sigma w_j G_j\) is hyperbolic with respect to \(\widetilde w\) with hyperbolicity cone equal to the semidefinite slice. The authors show readily that the converse holds for \(n= 2\).
A conjecture made by P. Lax in 1958 [Commun. Pure Appl. Math. 11, 175–194 (1958; Zbl 0086.01603)] proposes that all hyperbolic polynomials in 3 variables are likewise easily described in terms of determinants of symmetric matrices: A polynomial \(p\) on \(\mathbb{R}^3\) is hyperbolic of degree \(d\) with respect to the vector \(e= (1,0,0)\) and satisfies \(p(e)= 1\) if and only if there are matrices \(B,C \in\mathbb{S}^d\) such that \(p(x,y,z)\equiv\text{det}(xI+ yB+ zC)\). A polynomial \(q\) on \(\mathbb{R}^2\) is called a real zero polynomial if for every \((y,z)\in\mathbb{R}^2\) the univariate polynomial \(\widetilde q(t):= q(ty,tz)\) has all roots real. J. W. Helton and V. Vinnikov [Linear matrix inequality representation of sets. Technical report, Mathematics Department, UCSD (2002)] proved that a polynomial \(q\) on \(\mathbb{R}^2\) is a real zero polynomial of degree \(d\) satisfying \(q(0,0)= 1\) if and only if there are matrices \(B,C\in\mathbb{S}^d\) such that \(q(y,z)\equiv\text{det}(I+ yB+ zC)\). The authors show that this theorem is equivalent to the Lax conjecture.
The authors prove that a nonempty semidefinite slice is a hyperbolicity cone by showing that if the semidefinite slice contains the vector \(\widetilde w\), then the polynomial \(p(w):= \text{det\,}\Sigma w_j G_j\) is hyperbolic with respect to \(\widetilde w\) with hyperbolicity cone equal to the semidefinite slice. The authors show readily that the converse holds for \(n= 2\).
A conjecture made by P. Lax in 1958 [Commun. Pure Appl. Math. 11, 175–194 (1958; Zbl 0086.01603)] proposes that all hyperbolic polynomials in 3 variables are likewise easily described in terms of determinants of symmetric matrices: A polynomial \(p\) on \(\mathbb{R}^3\) is hyperbolic of degree \(d\) with respect to the vector \(e= (1,0,0)\) and satisfies \(p(e)= 1\) if and only if there are matrices \(B,C \in\mathbb{S}^d\) such that \(p(x,y,z)\equiv\text{det}(xI+ yB+ zC)\). A polynomial \(q\) on \(\mathbb{R}^2\) is called a real zero polynomial if for every \((y,z)\in\mathbb{R}^2\) the univariate polynomial \(\widetilde q(t):= q(ty,tz)\) has all roots real. J. W. Helton and V. Vinnikov [Linear matrix inequality representation of sets. Technical report, Mathematics Department, UCSD (2002)] proved that a polynomial \(q\) on \(\mathbb{R}^2\) is a real zero polynomial of degree \(d\) satisfying \(q(0,0)= 1\) if and only if there are matrices \(B,C\in\mathbb{S}^d\) such that \(q(y,z)\equiv\text{det}(I+ yB+ zC)\). The authors show that this theorem is equivalent to the Lax conjecture.
Reviewer: Rabe von Randow (Bonn)
MSC:
| 90C25 | Convex programming |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
Citations:
Zbl 0086.01603References:
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