We completely characterize non-trivial partition regularity of systems of affine algebraic plane curves over algebraically closed fields of characteristic 0 using two different methods. The first uses a combination of ultrafilters, nonstandard analysis and basic algebraic geometry; the second uses a known result about colorings related to functions with no fixed points. This provides one of the first examples of a large class of nonlinear equations whose partition regularity is completely characterized. Finally, we provide an explicit upper bound on the number of colors needed to disprove the partition regularity of those systems that are not non-trivially partition regular.

Partition regularity of affine algebraic plane curves / L. Luperi Baglini, P. Arruda. - (2022 Feb 09).

Partition regularity of affine algebraic plane curves

L. Luperi Baglini
Primo
;
2022-02-09

Abstract

We completely characterize non-trivial partition regularity of systems of affine algebraic plane curves over algebraically closed fields of characteristic 0 using two different methods. The first uses a combination of ultrafilters, nonstandard analysis and basic algebraic geometry; the second uses a known result about colorings related to functions with no fixed points. This provides one of the first examples of a large class of nonlinear equations whose partition regularity is completely characterized. Finally, we provide an explicit upper bound on the number of colors needed to disprove the partition regularity of those systems that are not non-trivially partition regular.
partition regularity; ultrafilters; nonstandard analysis; algebraic geometry
Settore MAT/01 - Logica Matematica
https://arxiv.org/pdf/2202.04535.pdf
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/906890
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