Deformations for Function Fields

Shape deformations based on vector fields

This thesis explores applications of vector field processing to shape deformations. We present a novel method to construct divergence-free vector fields which are used to deform shapes by vector field integration (Chapter 2). The resulting deformation is volume-preserving and no self-intersections occur. We add more controllability to this approach by introducing implicit boundaries (Chapter 3)...

Algorithms for Function Fields

Approximation Exponents for Function Fields

For diophantine approximation of algebraic real numbers by rationals, Roth’s celebrated theorem settles the issue of the approximation exponent completely in the number field case. But in spite of strong analogies between number fields and function fields, its naive analogue fails in function fields of finite characteristic, and the situation is not even conjecturally understood. In this short ...

Riemann Hypothesis for function fields

1 1 Preliminaries 1 1.1 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Primes and Divisors . . . . . . . . . . . . . . . . . . . . 2 1.2.2 The Picard Group . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Riemann-Roch . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Notation . . . . ...

Gauss Problem for Function Fields

According to a celebrated conjecture of Gauß , there are infinitely many real quadratic fields whose ring of integers is principal. We recall this conjecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solutions; namely we prove that there are infinite families of principal rings of algebraic functions ...