Decomposing multivariate polynomials with structured low-rank matrix completion

We are focused on numerical methods for decomposing a multivariate polynomial as a sum of univariate polynomials in linear forms. The main tool is the recent result on correspondence between the Waring rank of a homogeneous polynomial and the rank of a partially known quasi-Hankel matrix constructed from the coefficients of the polynomial. Based on this correspondence, we show that the original decomposition problem can be reformulated as structured low-rank matrix completion (or as structured low-rank approximation in the case of approximate decomposition). We construct algorithms for the polynomial decomposition problem. In the case of bivariate polynomials, we provide an extension of the wellknown Sylvester algorithm for binary forms.