Test sets for nonnegativity of polynomials invariant under a finite reflection group

Abstract

Abstract. A set S ⊂ R n is a nonnegativity witness for a set U of real homogeneous polynomials if F in U is nonnegative on R n if and only if it is nonnegative at all points of S. We prove that the union of the hyperplanes perpendicular to the elements of a root system Φ ⊆ R n is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by Φ. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by −1. We then characterize subspaces of forms of arbitrarily high degree where this union of hyperplanes is a nonnegativity witness set. Finally we propose a conjectural generalization of Timofte's half-degree principle for finite reflection groups. IntroductionThe general problem of deciding whether a homogeneous polynomial F in n variables with real coefficients is nonnegative has been one of the guiding questions of real algebraic geometry since the time of Hilbert. The intrinsic interest of this question has been complemented by a recent surge in its applications to polynomial optimization [1], control theory [9] and to the general theory of aproximation algorithms [14] among others. For all these applications, finding simple and efficient certificates for guaranteeing the nonnegativity of a given polynomial is of paramount importance.In the case of polynomials which are invariant under symmetric groups, such certificates have taken the form of "nonnegativity witness sets", that is subsets S ⊂ R n such that nonnegativity of F on S is equivalent to the nonnegativity of F in R n . If the witness set S is sufficiently simple then verifying that F is nonnegative becomes a simpler problem. ). An n-ary even symmetric sextic is nonnegative if and only if it is so at the n points (1, 0 . . . , 0), (1, 1, 0, . . . , 0), . . . , (1, 1 . . . , 1).2010 Mathematics Subject Classification. Primary 14P05 Secondary 20F55.

Reflection groups, reflection arrangements, and invariant real varieties

“…then the algebraic independence of the basic invariants implies that Theorem 11 can be applied to proof the following corollary. Under the assumption that f is homogeneous, the second part of the corollary recovers the main result of Acevedo and Velasco [1].…”

“…Example 1 also serves as a counterexample to generalizations of Corollary 6 to all finite reflection groups considered in [6, Lemma 1'] (without a proof) and [2, Statement 3.3]. Moreover, in the language of Acevedo and Velasco [1,Definition 7], it is the first example of a reflection group not satisfying the minor factorization condition.…”

“…Acevedo and Velasco [1] independently considered the related problem of certifying nonnegativity of G-invariant homogeneous polynomials. They show that low-degree forms (where the exact degree depends on the group) are nonnegative if and only if they are nonnegative on H n−1 (G).…”

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

“…Finally, a natural question is to explore the connections to invariant polynomials of other groups, most notably finite reflection groups. In [10,1] the authors showed that the image of polynomial functions invariant by a finite reflection group can be described by the points on flats in the hyperplane arrangement, if the degree is sufficiently small. We expect that the notions and techniques presented here can be transfered also to this more general setup.…”

“…1) + ⋅ ⋅ ⋅ + (−1) n z n is hyperbolic}= {z ∈ H T n − z 1 T n−1 + ⋅ ⋅ ⋅ + (−1)n z n has only non-negative roots} the set of even hyperbolic polynomials. Furthermore, we defineE k ∶= {z ∈ E T n − z 1 T n−1 + ⋅ ⋅ ⋅ + (−1) n z n hasat most k positive roots} and E L (a) ∶= E ∩ L −1 (a) and E k L (a) accordingly.…”