Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach

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In 2011, Neil Hindman proved that for every natural number n,m the polynomial equation* ∑ᵢ₌₁ⁿ xᵢ-∏ⱼ₌₁ᵐ yⱼ equation* has monochromatic solutions for every finite coloration of N. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials P(x₁,...,xₙ,y₁,...,yₘ) of the following kind: equation* P(x₁,...,xₙ,y₁,...,yₘ)=∑ᵢ₌₁ⁿaᵢxᵢMᵢ(y₁,...,yₘ), equation* where n,m are natural numbers, ∑ᵢ₌₁ⁿaᵢxᵢ has monochromatic solutions for every finite coloration of N and the degree of each variable y₁,...,yₘ in Mᵢ(y₁,...,yₘ) is at most one. An example of such a polynomial is equation* x₁y₁+x₂y₁y₂-x₃.equation* The second class of polynomials generalizing Hindman's result is more complicated to describe; its particularity is that the degree of some of the involved variables can be greater than one. The technique that we use relies on an approach to ultrafilters based on Nonstandard Analysis. Perhaps, the most interesting aspect of this technique is that, by carefully chosing the appropriate nonstandard setting, the proof of the main results can be obtained by very simple algebraic considerations.