Radical parametrizations of algebraic curves

It is well known that an algebraic curve can be parametrized by rational functions if and only if its genus is zero. The aim of this project is to study a more general class of curves: those parametrizable by radicals, that is, which admit a parametrization involving field operations and root extractions. This class is significantly larger than the previous one: it contains all elliptic and hyperelliptic functions, thus there are curves of every genus which have radical parametrizations. We approach the problem from an algorithmical point of view. In other words, our goal is to devise the best possible algorithms that decide whether a curve can be parametrized by radicals and compute one/allhe best radical parametrizations in the affirmative case. Naturally, this involves theoretical research into structural problems like a good definition of what is a better parametrization, what can we say about the structure of the class of all radical parametrization of a curve, etc. These questions go back to Zariski.

It is well-known that an algebraic curve can be parametrized by rational functions if and only if its genus is zero. The aim of this project was to study a more general class of curves: those parametrizable by radicals, that is, which admit a parametrization involving field operations and root extractions. This class is significantly larger than the previous one: it contains all elliptic and hyperelliptic functions, thus there are curves of every genus which have radical parametrizations.One way to construct radical parametrizations is to start with a k : 1 map to line, where the number k is as small as possible (this number is also called the gonality of the curve).