On diagonal cubic surfaces

Andrew Bremner

doi:10.1007/BF01258266

On diagonal cubic surfaces

Andrew Bremner

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We show that the affine surfaces x³+y³+cz³=c, c ∈Q, in the cases c≠2, c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface. However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.

Original language	English (US)
Pages (from-to)	21-32
Number of pages	12
Journal	Manuscripta Mathematica
Volume	62
Issue number	1
DOIs	https://doi.org/10.1007/BF01258266
State	Published - Mar 1988

ASJC Scopus subject areas

General Mathematics

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10.1007/BF01258266

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title = "On diagonal cubic surfaces",

abstract = "We show that the affine surfaces x3+y3+cz3=c, c ∈Q, in the cases c≠2, c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface. However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.",

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N2 - We show that the affine surfaces x3+y3+cz3=c, c ∈Q, in the cases c≠2, c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface. However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.

AB - We show that the affine surfaces x3+y3+cz3=c, c ∈Q, in the cases c≠2, c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface. However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.

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On diagonal cubic surfaces

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