Rado functionals and applications

Abstract: We study Rado functionals and the maximal condition (first introduced by J. M. Barret et al.) in terms of the partition regularity of mixed systems of linear equations and inequalities. By strengthening the maximal Rado condition, we provide a sufficient condition for the partition regularity of polynomial equations over some infinite subsets of a given commutative ring. By applying these results, we derive an extension of a previous result obtained by M. Di Nasso and L. Luperi Baglini concerning partition regular inhomogeneous polynomials in three variables and also conditions for the partition regularity of equations of the form $H(xz^\rho,y)=0$, where $\rho$ is a non-zero rational and $H\in\mathbb{Z}[x,y]$ is a homogeneous polynomial.