On algebraic curves A(x) - B(y) = 0 of genus zero

Fedor Pakovich

doi:10.1007/s00209-017-1889-9

On algebraic curves A(x) - B(y) = 0 of genus zero

Fedor Pakovich

Department of Mathematics

Ben-Gurion University of the Negev

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

Abstract

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form EA,B:A(x)-B(y)=0, where A, B∈ C(z). We also investigate “series” of curves E_A _, _B of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions B_i, such that deg B_i→ ∞ and all the curves A(x) - B_i(y) = 0 are irreducible and have genus zero, exists if and only if the Galois closure of the field extension C(z) / C(A) has genus zero or one.

Original language	American English
Pages (from-to)	299-310
Number of pages	12
Journal	Mathematische Zeitschrift
Volume	288
Issue number	1-2
DOIs	https://doi.org/10.1007/s00209-017-1889-9
State	Published - 1 Feb 2018

Keywords

Galois coverings
Rational points
Separated variable polynomials
Two-dimensional orbifolds

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00209-017-1889-9

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abstract = "Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form EA,B:A(x)-B(y)=0, where A, B∈ C(z). We also investigate “series” of curves EA , B of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions Bi, such that deg Bi→ ∞ and all the curves A(x) - Bi(y) = 0 are irreducible and have genus zero, exists if and only if the Galois closure of the field extension C(z) / C(A) has genus zero or one.",

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AB - Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form EA,B:A(x)-B(y)=0, where A, B∈ C(z). We also investigate “series” of curves EA , B of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions Bi, such that deg Bi→ ∞ and all the curves A(x) - Bi(y) = 0 are irreducible and have genus zero, exists if and only if the Galois closure of the field extension C(z) / C(A) has genus zero or one.

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KW - Rational points

KW - Separated variable polynomials

KW - Two-dimensional orbifolds

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JF - Mathematische Zeitschrift

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On algebraic curves A(x) - B(y) = 0 of genus zero

Abstract

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