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  • 1806.05002

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Rado's criterion over squares and higher powers

Research output: Contribution to Journal/MagazineJournal articlepeer-review

<mark>Journal publication date</mark>16/02/2021
<mark>Journal</mark>Journal of the European Mathematical Society
Issue number6
Number of pages73
Pages (from-to)1925-1997
Publication StatusPublished
<mark>Original language</mark>English


We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive kth powers, provided the equation has at least (1+o(1))klogk variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado's theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers.