The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

Abstract

In this paper, we conjecture a formula for the value of the Pythagoras number for real multivariate sum of squares polynomials as a function of the (total or coordinate) degree and the number of variables. The conjecture is based on the comparison between the number of parameters and the number of conditions for a corresponding low-rank representation. This is then numerically verified for a number of examples. Additionally, we discuss the Pythagoras number of (complex) multivariate Laurent polynomials that are sum of square magnitudes of polynomials on the n-torus. For both types of polynomials, we also propose an algorithm to numerically compute the Pythagoras number and give some numerical illustrations.

Algebraic Stories from One and from the Other Pockets

“…Exact values of d k,n , and the behaviour of rk k (kd, n) for d ≤ d k,n , have also been computed in the case k = 2 for n = 3, 4; see (Lundqvist et al (2017), Appendix). These results agree with the following conjecture suggested to the authors by Ottaviani in 2014 in a private communication (in the case k = 2, this conjecture agrees with ( [Le et al (2013), Conjecture 1])).…”

On bounds of the Pythagoras number of the sum of square magnitudes of Laurent polynomials

“…Note that the conjectured formula of the Pythagoras number of sosm-polynomials given in [13] satisfies the bounds of Theorem 2. The upper bound turns out to be a sharp one.…”

Low rank solutions to differentiable systems over matrices and applications

“…RM-problem ( 1) is computationally NP-hard in general, even when C is an affine subset of R m×n . There hence is a number of algorithms for solving this problem with respect to special cases of C, see, eg., [9,16,24] and the references there in. When the constraints are defined by linear matrix equations, i.e., C is the solution set of a linear system of equations ℓ(X) = b ∈ R k , the present problem is called affine rank minimization problem (shortly, ARM-problem) and is in the form [24] minimize rank(X) subject to X ∈ R m×n , ℓ(X) = b.…”