Symmetric and Even Symmetric forms: Test sets for Positivity and Sum of Squares representations

Symmetric and Even Symmetric forms: Test sets for Positivity and Sum of Squares representations

Sums of squares representations of polynomials are of fundamental importance in real algebraic geometry. In 1888, Hilbert gave a complete characterisation of the pairs (n, 2d) for which a n-ary 2d-ic form non-negative on R n can be written as sums of squares of other forms, namely P n,2d = Σn,2d if and only if n = 2, d = 1, or (n, 2d) = (3, 4), where P n,2d and Σn,2d are respectively the cones of positive semidefinite (psd) and sum of squares (sos) forms (real homogenous polynomials) of degree 2d in n variables. In 1976 Choi and Lam gave an analogue of Hilberts 1888 Theorem for symmetric forms. By assuming the existence of psd not sos symmetric n-ary quartics for n ≥ 5, they showed that this characterisation remains unchanged for symmetric forms.
In this talk we will first present test sets for positivity of symmetric and even symmetric forms given by Choi, Lam, Reznick, Harris and Timofte. We then complete the above assertion of Choi-Lam and present our analogue of Hilbert’s characterisation under the additional assumptions of even symmetry on the given form. We establish that an even symmetric n-ary 2d-ic psd form is sos if and only if n = 2 or d = 1 or (n, 2d) = (n, 4)n≥3 or (n, 2d) = (3, 8). This is joint work with S. Kuhlmann and B. Reznick.