Reflection groups and cones of sums of squares. (English) Zbl 1521.14102
Studying the difference between the Cone of Sums of Squares and the Cone of Nonnegative Real Polynomials is a crucial problem in real algebraic geometry. In this article, the authors focus extensively on the structure of the Sums of Squares Cone, which remains invariant under reflection groups. In particular, they effectively analyze the structure of SoS cones invariant under \(A_n\), \(B_n\), and \(D_n\), utilizing Specht’s polynomial representations of reflection group as introduced by H. Morita and H. F. Yamada [Hokkaido Math. J. 27, No. 3, 505–515 (1998; Zbl 0919.20027)] in their work on Higher Specht Polynomials for the complex reflection group.
Research on when non-negative polynomials can be expressed as sums of squares has a long history, dating back to Hilbert’s work on forms with given degrees and the number of variables. Notably, studies have been conducted on symmetric forms, such as Choi, Lam, and Resnick’s results on symmetric sextics and Harris’s investigations into symmetric ternary octics, where it has been established that non-negative polynomials can indeed be represented as sums of squares.
Extensive research in the invariant theory of finite reflection groups has allowed the generalization of the theories and results from symmetric polynomials. In particular, this paper, focusing on reflection groups that are invariant under \(A_n\), \(B_n\), and \(D_n\), achieves a concrete characterization of the cone of invariant sums of squares in Theorem 3.11.
Using this characterization, in Theorem 4.1, the authors provide an explicit description of the dual cone of even symmetric ternary octic sums of squares, essentially reaffirming Harris’s results concerning symmetric ternary octics. Furthermore, in Theorem 4.18, they prove that any nonnegative ternary octics invariant under \(D_3\) can be expressed as a sum of squares. In Theorem 4.22, the authors shed light on the structure of the dual cone of sums of squares of quaternary quartics invariant under \(D_4\), and in Theorem 4.26, they establish whether all nonnegative polynomials for a given \((n,2d)\) are sums of squares invariant under \(D_n\). Finally, the article concludes by highlighting some connections to non-negativity testing of forms using semidefinite programming in the last subsection.
Research on when non-negative polynomials can be expressed as sums of squares has a long history, dating back to Hilbert’s work on forms with given degrees and the number of variables. Notably, studies have been conducted on symmetric forms, such as Choi, Lam, and Resnick’s results on symmetric sextics and Harris’s investigations into symmetric ternary octics, where it has been established that non-negative polynomials can indeed be represented as sums of squares.
Extensive research in the invariant theory of finite reflection groups has allowed the generalization of the theories and results from symmetric polynomials. In particular, this paper, focusing on reflection groups that are invariant under \(A_n\), \(B_n\), and \(D_n\), achieves a concrete characterization of the cone of invariant sums of squares in Theorem 3.11.
Using this characterization, in Theorem 4.1, the authors provide an explicit description of the dual cone of even symmetric ternary octic sums of squares, essentially reaffirming Harris’s results concerning symmetric ternary octics. Furthermore, in Theorem 4.18, they prove that any nonnegative ternary octics invariant under \(D_3\) can be expressed as a sum of squares. In Theorem 4.22, the authors shed light on the structure of the dual cone of sums of squares of quaternary quartics invariant under \(D_4\), and in Theorem 4.26, they establish whether all nonnegative polynomials for a given \((n,2d)\) are sums of squares invariant under \(D_n\). Finally, the article concludes by highlighting some connections to non-negativity testing of forms using semidefinite programming in the last subsection.
Reviewer: Jaewoo Jung (Daejeon)
MSC:
| 14P99 | Real algebraic and real-analytic geometry |
| 05E05 | Symmetric functions and generalizations |
| 90C22 | Semidefinite programming |
Keywords:
sums of squares; reflection groups; invariant theory; symmetry reduction; real algebraic geometry; polynomial optimizationCitations:
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