It is one of the main questions in real algebra whether a positive semidefinite (psd) polynomial is a sum of squares (sos) of polynomials. By Hilbert, not every psd form is sos. The first explicit example was given by Motzkin. A result of Hurwitz states that every form \(a_1x_1^{2d}+\ldots+a_nx_n^{2d}-2dx_1^{a_1}\cdots x_n^{a_n}\), where \(a_i\in\mathbb{Z}_{\geq 0}\) have sum \(2d\), is sos.
The authors of the paper under review extend Hurwitz’s theorem. They introduce the notion of a diagonal minus tail (dmt) form. This is a form \(F(x)=D(x)-T(x)\) where the diagonal part \(D(x)\) has the shape \(D(x)=\sum_{i=1}^nb_ix_i^{2d}\) with \(d\in\mathbb{Z}_{\geq 1}\) and \(b_i\geq 0\), and the tail \(T(x)\) has the shape \(T(x)=\sum_{i\in I}a_ix^i\) with \(a_i\geq 0\) and \(I\subseteq\{i=(i_1,\ldots,i_n)\in\mathbb{Z}^n_{\geq 0}: i_1,\ldots,i_n\leq 2d-1, i_1+\ldots+i_n=2d\}\). The main result of the article states that every psd dmt form is a sum of binomial and monomial squares (sbs). This is done in an algorithmic way.