Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach

In 2011, Neil Hindman proved that for every natural number $n,m$ the polynomial \begin{equation*} \sum_{i=1}^{n} x_{i}-\prod\limits_{j=1}^{m} y_{j} \end{equation*} has monochromatic solutions for every finite coloration of $\mathbb{N}$. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials $P(x_{1},...,x_{n},y_{1},...,y_{m})$ of the following kind: \begin{equation*} P(x_{1},...,x_{n},y_{1},...,y_{m})=\sum_{i=1}^{n}a_{i}x_{i}M_{i}(y_{1},...,y_{m}), \end{equation*} where $n,m$ are natural numbers, $\sum\limits_{i=1}^{n}a_{i}x_{i}$ has monochromatic solutions for every finite coloration of $\mathbb{N}$ and the degree of each variable $y_{1},...,y_{m}$ in $M_{i}(y_{1},...,y_{m})$ is at most one. An example of such a polynomial is \begin{equation*} x_{1}y_{1}+x_{2}y_{1}y_{2}-x_{3}.\end{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

A nonstandard technique in combinatorial number theory

In [9], [15] it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we present three applications of this technique: the first one is a new proof of a known result regarding the algebra of \betaN, namely that the center of the semigroup (\beta\mathbb{N};\oplus) is \mathbb{N}; the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of lenght three; the third one regards the partition regular polynomials in Z[X], namely the polynomials in Z[X] that have a monochromatic solution for every finite coloration of N. We will study this last application in more detail: we will prove some algebraical properties of the sets of such polynomials and we will present a few examples of nonlinear partition regular polynomials. In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis

Ultrafilters maximal for finite embeddability

In [1] the authors showed some basic properties of a pre-order that arose in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and they presented its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-\v{C}ech compactification of the discrete space of natural numbers. In this present paper we continue the study of these pre-orders. In particular, we prove that there exist ultrafilters maximal for finite embeddability, and we show that the set of such ultrafilters is the closure of the minimal bilateral ideal in the semigroup (\bN,\oplus), namely \overline{K(\bN,\oplus)}. As a consequence, we easily derive many combinatorial properties of ultrafilters in \overline{K(\bN,\oplus)}. We also give an alternative proof of our main result based on nonstandard models of arithmetic

Generalized Functions Beyond Distributions

Ultrafunctions are a particular class of functions defined on a Non Archimedean field R^{*}\supset R. They have been introduced and studied in some previous works ([1],[2],[3]). In this paper we introduce a modified notion of ultrafunction and we discuss sistematically the properties that this modification allows. In particular, we will concentrated on the definition and the properties of the operators of derivation and integration of ultrafunctions.Comment: 29 pages, Keywords: Ultrafunctions, Delta function, distributions, Non Archimedean Mathematics, Non Standard Analysis. arXiv admin note: text overlap with arXiv:1302.715

Basic properties of ultrafunctions

Ultrafunctions are a particular class of functions defined on a non-Archimedean field. They provide generalized solutions to functional equations which do not have any solutions among the real functions or the distributions. In this paper we analyze sistematically some basic properties of the spaces of ultrafunctions.Comment: 25 page