Avinash Kulkarni, MPI Leipzig: "Approximate solutions of zero-dimensional polynomial systems over Q_p", and "The arithmetic of uniquely trigonal genus 4 curves"

Referent: Avinash Kulkarni, MPI Leipzig

Zeit: Tuesday,  05.02.2019, 15:30 h

Ort:Raum 48-436

Abstract:

1) "Approximate solutions of zero-dimensional polynomial systems over Q_p"

In this talk, we look at the details of a truncated normal form based method for computing the (approximate) solutions to a zero-dimensional polynomial system. In particular, we discuss how the algorithm of van Baren, Mourrain, and Telen for solving systems over the real and complex fields can be adapted to the p-adic setting.

2) "The arithmetic of uniquely trigonal genus 4 curves"

A uniquely trigonal curve is a smooth algebraic curve that has an essentially unique morphism to P_1 of degree 3. The uniquely trigonal curves of genus 4 are closely related to del Pezzo surfaces of degree one. In this talk, we give two examples of how this connection can be used to generate interesting results in number theory. The first result pertains to class groups of cubic fields; we construct an infinite family of cubic number fields whose class groups have many 2-torsion elements. For the second application, we consider the uniquely trigonal genus 4 curves from the perspective of arithmetic invariant theory.

The first talk is in the "spirit" of the seminar, finding points on varieties, while the second representes work he has done in his PhD.