Advanced Studies in Pure Mathematics

Some geometric-arithmetic aspects of separated variable curves

Viet Khac Nguyen

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The paper shows certain geometric-arithmetic aspects with some new observations in consideration of variables separated curves, e.g. relationships with generalized Chebyshev polynomials, Chebyshev pencils, local variant at infinity of Stothers–Mason $abc$-theorem, Stothers–Langevin pairs, Pell–Abel conics (or polynomial Pell equations), Belyi maps, etc. Several case studies and open problems are discussed.

Article information

Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 161-172

Received: 22 March 2012
Revised: 30 July 2012
First available in Project Euclid: 19 October 2018

Permanent link to this document euclid.aspm/1539916286

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25] 14H10: Families, moduli (algebraic)

Chebyshev polynomial Stothers–Mason abc-theorem Stothers–Langevin pair Pell–Abel conics


Nguyen, Viet Khac. Some geometric-arithmetic aspects of separated variable curves. Singularities in Geometry and Topology 2011, 161--172, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06610161.

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