OPTIMAL REPARAMETRIZATION OF POLYNOMIAL ALGEBRAIC CURVES

Abstract

In this paper, we present an algorithm for optimally parametrizing polynomial algebraic curves. Let [Formula: see text] be a polynomial plane algebraic curve given by a polynomial parametrization [Formula: see text] , where [Formula: see text] is a finite field extension of a field [Formula: see text] of characteristic zero. We prove that if [Formula: see text] is polynomial over [Formula: see text] , then Weil's descente variety associated with [Formula: see text] is surprisingly simple; it is, in fact, a line. Applying this result we are able to derive an effective algorithm to algebraically optimal reparametrize polynomial algebraic curves.