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

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].


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