A limiting result for the partition regularity of functional equations – Paulo Arruda (Universität Wien)


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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