Jonathan Chapman

Partition and density regularity for diagonal systems of equations

A system of polynomial equations is called partition regular if every finite colouring of the positive integers produces monochromatic solutions to the system. A system is called density regular if it has solutions over every set of integers with positive upper density. A classical theorem of Rado characterises partition regularity for linear systems, whilst Szemerédi’s theorem classifies all density regular linear systems. In this talk, I will report on my recent work on classifying partition and density regularity for sufficiently non-singular systems of diagonal polynomial equations.