Smooth hyperbolicity cones are spectrahedral shadows
نویسندگان
چکیده
Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow.
منابع مشابه
Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones
We give explicit polynomial-sized (in n and k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k in n variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we const...
متن کاملExponential lower bounds on spectrahedral representations of hyperbolicity cones
The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree d in n variables contains (n/d) pairwise distant cones in the Hausdorff metric, and therefore that any semidefinite representation of such polynomials must have dimension at lea...
متن کاملHyperbolic Polynomials and Generalized Clifford Algebras
We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if −1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, t...
متن کاملUnbounded Convex Semialgebraic Sets as Spectrahedral Shadows
Recently, Helton and Nie [3] showed that a compact convex semialgebraic set S is a spectrahedral shadow if the boundary of S is nonsingular and has positive curvature. In this paper, we generalize their result to unbounded sets, and also study the effect of the perspective transform on singularities.
متن کاملLecture 11 : Hyperbolic Polynomials and Hyperbolicity Cones
In this lecture, we will introduce the concept of hyperbolic polynomials, a generalization of real stable polynomials. We will also introduce the concept of hyperbolicity cones, which are the set of directions along which a polynomial is always real-rooted. We will prove that hyperbolicity cones are convex, study some of their properties, and invyestigate the connection between barrier argument...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Math. Program.
دوره 153 شماره
صفحات -
تاریخ انتشار 2015