Regularity of certain diophantine equations

Abstract

In Ramsey theory, there is a vast literature on regularity questions of linear diophantine equations. Some problems in higher degree have been considered recently. Here, we show that, for every pair of positive integers r and n, there exists an integer $$B=B(r)$$ such that the diophantine equation $$\begin{aligned}&\prod _{m=1}^{n}\left( \sum _{i=1}^{k_m} a_{m,i} x_{m,i} - \sum _{j=1}^{l_m}b_{m,j}y_{m,j}\right) = B \end{aligned}$$ with $$\begin{aligned}&\sum _{i=1}^{k_m} a_{m,i} = \sum _{j=1}^{l_m}b_{m,j} \qquad \forall m = 1,\ldots , n \end{aligned}$$ is r-regular, where $$ k_m$$ , $$l_m$$ are also positive integers and $$a_{m,i}, b_{m,j}$$ are non-zero integers.