Rado functionals and applications

We study Rado functionals and the maximal Rado condition (first introduced
by Barret, Lupini, and Moreira in 2021) 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 integral domain. By applying these results, we
derive an extension of a previous result obtained by Di Nasso and Luperi
Baglini concerning partition regular inhomogeneous polynomials in three
variables and also conditions for the partition regularity of equations of
the form H(xzρ, y) = 0, where ρ is a non-zero rational and H ∈ Z[x, y]
is a homogeneous polynomial.