A property of quasi-diagonal forms (Q1940073)

The aim of the paper is to prove the following result: Theorem: Let \(k\) be a positive integer and let \(F_i\in \mathbb{Z}[\mathbf{x}_i]\) be forms of degree \(k\) in disjoint vectors of variables such that either all forms \(F_i\) are non-singular or not all forms \(F_i\) are semidefinite of the same sign. Then there exists a positive integer \(s_0\) such that, for every \(s\), all integers represented by \(\sum_{i=1}^sF_i(\mathbf{x}_i)\) over \(\mathbb{Z}\) are represented by \(\sum_{i=1}^{s_0}F_i(\mathbf{x}_i)\) over \(\mathbb{Z}\).