  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.
