Partition Regularity for Systems of Diagonal Equations

We consider systems of $n$ diagonal equations in $k$th powers. Our main result shows that if the coefficient matrix of such a system is sufficiently nonsingular, then the system is partition regular if and only if it satisfies Rado’s columns condition. Furthermore, if the system also admits constant solutions, then we prove that the system has nontrivial solutions over every set of integers of positive upper density.