We study genus 3 hyperelliptic curves which have an extra involution. The locus L 3 of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli H 3 . We find a birational parametrization of this locus by affine 3-space. For every moduli point p ∈ H 3 such that |Aut(p)| > 2, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points p ∈ H 3 such that |Aut(p)| > 4. This is the first time that such a rational model of these curves appears in the literature.2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

In this paper we discuss several notions of decomposition for multivariate rational functions, and we present algorithms for decomposing multivariate rational functions over an arbitrary field. We also provide a very efficient method to decide if a unirational field has transcendence degree one, and in the affirmative case to compute the generator.

In this paper we present an algorithm to compute all unirational fields of transcendence degree one containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate rational function f of the form f = g(h), where g is a univariate rational function and h a multivariate one.

We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.

In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We provide algorithms to implicitize radical parametrizations and to check whether a radical parametrization can be reparametrized into a rational parametrization.Comment: 26 pages; revised version accepted for publication in Computer Aided Geometric Desig