## Advanced Studies in Pure Mathematics

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

### Some geometric-arithmetic aspects of separated variable curves

#### Abstract

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

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

**Dates**

Received: 22 March 2012

Revised: 30 July 2012

First available in Project Euclid:
19 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1539916286

**Digital Object Identifier**

doi:10.2969/aspm/06610161

**Mathematical Reviews number (MathSciNet)**

MR3382049

**Zentralblatt MATH identifier**

1360.11076

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aspm/1539916286