PARAMETRIZATION OF ALGEBRAIC-CURVES OVER OPTIMAL FIELD-EXTENSIONS

In this paper we investigate the problem of determining rational param etrizations of plane algebraic curves over an algebraic extension of l east degree over the field of definition. This problem reduces to the problem of finding simple points with coordinates in the field of defi nition on algebraic curves of genus 0. Consequently we are also able t o decide parametrizability over the reals. We generalize a classical t heorem of Hilbert and Hurwitz about birational transformations. An eff icient algorithm for computing such optimal parametrizations is presen ted. (C) 1997 Academic Press Limited.