Reduction of binary forms via the hyperbolic centroid

Artur Elezi, Tanush Shaska

I-AIMS paper 2017-1 · Published in Lobachevskii J. Math. 42 (2021), 84–95 · 2021

In this paper we introduce a reduction theory based on the hyperbolic center of mass, which is different from the reduction introduced by Julia (1917). We show that the zero map via the Julia quadratic is different than the hyperbolic center of mass. Moreover, we discover some interesting formulas for computing the hyperbolic centroid.

MSC classes: Primary 51M10, 11E16; Secondary 51M15, 14D20

PDF

References

1. L. Beshaj, Minimal weierstrass equations for genus 2 curves (2016), available at 1612.08318.
2. L. Beshaj, R. Hidalgo, A. Malmendier, S. Kruk, S. Quispe, and T. Shaska, Rational points on the moduli space of genus two, Contemporary Mathematics (AMS) (2018).
3. Lubjana Beshaj, Reduction theory of binary forms, Advances on superelliptic curves and their applications, 2015, pp. 84–116.
4. John E. Cremona, Reduction of binary cubic and quartic forms, LMS J. Comput Math 2 (1999), 64–94.
5. G. A. Galperin, A concept of the mass center of a system of material points in the constant curvature spaces, Comm. Math. Phys. 154 (1993), no. 1, 63–84.
6. Gaston Julia, Etude sur les formes binaires non quadratiques `á indéterminées réelles ou complexes., Mémories de lAcadémie des Sciences de lInsitut de France 55 (1917), 1–296.
7. T. Shaska and L. Beshaj, Heights on algebraic curves, Advances on superelliptic curves and their applications, 2015, pp. 137–175.
8. Michael Stoll and John E. Cremona, On the reduction theory of binary forms, J. Reine Angew. Math. 565 (2003), 79–99. (2005e:11091)

All preprints