This thesis is devoted to a study of real holomorphy rings in R(x) and elliptic function fields R(x)( √ f(x)), f is a monic polynomial of degree 3 with no multiple roots in C over R and sums of powers in R(x). The objects of study are real valuation rings, real holomorphy rings and real places in the rational function fields R(x) and then investigate their extension to elliptic function fields ...

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid 1980s and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x, y) that are suitable for this framework. We then perform such computations for i...

A conjecture on vanishing and non-negativity of intersection multiplicity on the blow-up of a regular local ring at its closed point has been proposed. Proofs of vanishing, several special cases of non-negativity and a sufficient condition for non-negativity of this conjecture are described. The topic on intersection multiplicity on vector bundles is also addressed. Introduction. Let X be a reg...