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

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.