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Counting Monochromatic Solutions to Diagonal Diophantine Equations

Research output: Contribution to Journal/Magazine › Editorial › peer-review

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Counting Monochromatic Solutions to Diagonal Diophantine Equations. / Prendiville, Sean.
In: Discrete Analysis, Vol. 2021, 14, 17.09.2021.

Research output: Contribution to Journal/Magazine › Editorial › peer-review

Harvard

Prendiville, S 2021, 'Counting Monochromatic Solutions to Diagonal Diophantine Equations', Discrete Analysis, vol. 2021, 14. https://doi.org/10.19086/da.28173

APA

Prendiville, S. (2021). Counting Monochromatic Solutions to Diagonal Diophantine Equations. Discrete Analysis, 2021, Article 14. https://doi.org/10.19086/da.28173

Vancouver

Prendiville S. Counting Monochromatic Solutions to Diagonal Diophantine Equations. Discrete Analysis. 2021 Sept 17;2021:14. doi: 10.19086/da.28173

Author

Prendiville, Sean. / Counting Monochromatic Solutions to Diagonal Diophantine Equations. In: Discrete Analysis. 2021 ; Vol. 2021.

Bibtex

@article{6fa69326ae864f8ea0ff549f4c336218,
title = "Counting Monochromatic Solutions to Diagonal Diophantine Equations",
abstract = "We show how to adapt the Hardy–Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations obtained by setting a diagonal quadratic form equal to a linear form. As a consequence, we determine an algebraic criterion for when such equations are partition regular. Our methods involve discrete harmonic analysis and require a number of {\textquoteleft}mixed{\textquoteright} restriction estimates, which may be of independent interest.",
author = "Sean Prendiville",
year = "2021",
month = sep,
day = "17",
doi = "10.19086/da.28173",
language = "English",
volume = "2021",
journal = "Discrete Analysis",
issn = "2397-3129",
publisher = "Alliance of Diamond Open Access Journals",

}

RIS

TY - JOUR

T1 - Counting Monochromatic Solutions to Diagonal Diophantine Equations

AU - Prendiville, Sean

PY - 2021/9/17

Y1 - 2021/9/17

N2 - We show how to adapt the Hardy–Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations obtained by setting a diagonal quadratic form equal to a linear form. As a consequence, we determine an algebraic criterion for when such equations are partition regular. Our methods involve discrete harmonic analysis and require a number of ‘mixed’ restriction estimates, which may be of independent interest.

AB - We show how to adapt the Hardy–Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations obtained by setting a diagonal quadratic form equal to a linear form. As a consequence, we determine an algebraic criterion for when such equations are partition regular. Our methods involve discrete harmonic analysis and require a number of ‘mixed’ restriction estimates, which may be of independent interest.

U2 - 10.19086/da.28173

DO - 10.19086/da.28173

M3 - Editorial

AN - SCOPUS:85124641807

VL - 2021

JO - Discrete Analysis

JF - Discrete Analysis

SN - 2397-3129

M1 - 14

ER -