Limits of quotients of polynomial functions of three variables, Classification of G-graded twisted algebras and the computation of the F-rational locus

Abstract

This thesis is divided in three main parts. In the first part we provide a theoretical method to determine the existence of the limit of a quotient of polynomial functions of three variables. An algorithm to compute such limits in the case where the polynomials have rational coeffcients, or more generally, coefficients in a real finite extension of the rational numbers is also described. In the second part, for any finite abelian group G, we present an exact formula to count the G graded twisted algebras satisfying certain symmetry condition. Finally, in the third part we describe an algorithm to compute the F-rational locus of an affine algebra over a field of prime characteristic p 0 by computing first its global test ideal. As a consequence we deduce the Openness of the F-rational locus, a result originally proved in [27]