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On Rado conditions for nonlinear Diophantine equations

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Barrett, Jordan Mitchell, Lupini, Martino and Moreira, Joel (2021) On Rado conditions for nonlinear Diophantine equations. European Journal of Combinatorics, 94 . 103277. doi:10.1016/j.ejc.2020.103277

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Official URL: https://doi.org/10.1016/j.ejc.2020.103277

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Abstract

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado’s characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado’s theorem.

We verify that such a conjecture holds for the equations and for such that or . To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in such as , building on the recent result on the partition regularity of .

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Diophantine equations , Polynomials
Journal or Publication Title: European Journal of Combinatorics
Publisher: Academic Press
ISSN: 0195-6698
Official Date: May 2021
Dates:
DateEvent
May 2021Published
17 December 2020Available
10 August 2020Accepted
Volume: 94
Article Number: 103277
DOI: 10.1016/j.ejc.2020.103277
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIEDCalifornia Institute of Technologyhttp://dx.doi.org/10.13039/100006961
UNSPECIFIEDVictoria University of Wellingtonhttp://dx.doi.org/10.13039/501100001538
DMS-1600186National Science Foundationhttp://dx.doi.org/10.13039/501100008982
UNSPECIFIEDRoyal Society of New Zealandhttp://dx.doi.org/10.13039/501100001509
DMS-1700147National Science Foundationhttp://dx.doi.org/10.13039/501100008982
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