Abstract
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.
Highlights
Throughout the paper let R[X1, . . . , Xn] denote the ring of polynomials in n real variables and Hn,k the set of homogeneous polynomials of degree k in R[X1, . . . , Xn]
We study the case of forms in n variables of degree 2d that are symmetric, i.e., invariant under the action of the symmetric group Sn that permutes the variables
In this article we study the relationship between symmetric sums of squares and symmetric non-negative forms
Summary
Throughout the paper let R[X1, . . . , Xn] denote the ring of polynomials in n real variables and Hn,k the set of homogeneous polynomials (forms) of degree k in R[X1, . . . , Xn]. Choi and Lam [7] showed that the following symmetric form of degree 4 in 4 variables is non-negative but cannot be written as a sum of squares: X. In [3] the first author added to the work of Hilbert by showing that the gap between sum of squares and non-negative forms of fixed degree grows infinitely large with the number of variables if the degree is at least 4. This result has been recently refined by Ergur to the multihomogeneous case [10].
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