Rational solutions of pairs of diagonal equations, one cubic and one quadratic

Trevor D Wooley

doi:10.1112/plms/pdu054

Rational solutions of pairs of diagonal equations, one cubic and one quadratic

Trevor D Wooley

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument αx³+βx². Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10.

Original language	English
Pages (from-to)	325-356
Number of pages	32
Journal	Proceedings of the London Mathematical Society
Volume	110
Issue number	2
Early online date	21 Nov 2014
DOIs	https://doi.org/10.1112/plms/pdu054
Publication status	Published - Feb 2015

Keywords

math.NT
11D72
11L15
11L07
11P55
Weyl sums
Hardy-Littlewood method
Diophantine equations

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10.1112/plms/pdu054

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This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Oxford University Press at http://plms.oxfordjournals.org/content/110/2/325. Please refer to any applicable terms of use of the publisher.
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abstract = "We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument αx3+βx2. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10.",

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AU - Wooley, Trevor D

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N2 - We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument αx3+βx2. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10.

AB - We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument αx3+βx2. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10.

KW - math.NT

KW - 11D72

KW - 11L15

KW - 11L07

KW - 11P55

KW - Weyl sums

KW - Hardy-Littlewood method

KW - Diophantine equations

U2 - 10.1112/plms/pdu054

DO - 10.1112/plms/pdu054

M3 - Article (Academic Journal)

SN - 0024-6115

VL - 110

SP - 325

EP - 356

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 2

ER -

Rational solutions of pairs of diagonal equations, one cubic and one quadratic

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Keywords

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