Some geometric-arithmetic aspects of separated variable curves. (English) Zbl 1360.11076

Nguyen, Viet Khac

Blanlœil, Vincent (ed.) et al., Singularities in geometry and topology 2011. Proceedings of the 6th Franco-Japanese symposium on singularities, Fukuoka, Japan, September 5–10, 2011. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-026-6/hbk). Advanced Studies in Pure Mathematics 66, 161-172 (2015).

Summary: 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.
For the entire collection see [Zbl 1320.14001].

MSC:

11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H10	Families, moduli of curves (algebraic)
14H50	Plane and space curves

Keywords:

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