Dipartimenti di Matematica, Aula Seminari.
Our main result provides a condition under which the partition regularity (PR) of systems of functional equations over a given infinite set can be completely characterised by the existence of constant solutions. From this result, we characterise the PR of several families of non-linear equations over infinite subsets of $\mathbb C$, such as $\mathbb N$, $\mathbb Z$ or finitely generated multiplicative subgroups of $\mathbb C^\times$. Examples of applications of our main theorem includes:
1. A complete characterisation of the PR system polynomial equations in two variables over $\mathbb N$;
2. a complete characterisation of the PR of polyexponential equations over $\mathbb Z$;
3. a complete characterization of the PR of Thue-Mahler equations over $\mathbb Z$; and
4. the failure of Rado’s Theorem for linear equations over finitely generated subgroups of $\mathbb C^\times$.
Joint work with Lorenzo Luperi Baglini.
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