Abstract. In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.

This note presents a new and elementary proof of a statement that was first proved by Timofte [15]. It says that a symmetric real polynomial F of degree d in n variables is positive on R n ( on R n + ) if and only if it is so on the subset of points with at most max{⌊d/2⌋, 2} distinct components. The key idea of our new proof lies in the representation of the orbit space. The fact that for the case of the symmetric group S n it can be viewed as the set of normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way.

Abstract. Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of R k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleȋnik and Petrovskiȋ, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of R k . We obtain several asymptotically tight upper bounds. In particular, we prove that if S ⊂ R k is a semi-algebraic subset defined by a finite set of s symmetric polynomials of degree at most d, then the sum of the S k -equivariant Betti numbers of S with coefficients in Q is bounded by (skd) O(d) . Unlike the classical bounds on the ordinary Betti numbers of real algebraic varieties and semi-algebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.

We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables n and degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of n-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree 2d is fixed and the number of variables n grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric non-negative forms and sums of squares does not grow arbitrarily large for any fixed degree 2d. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree 4 the cones of non-negative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree 2d.

No abstract