David Sevilla

On Multivariate Rational Function Decomposition

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.

Radical parametrizations of algebraic curves by adjoint curves

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less than or equal to 4 and the curve is defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree d having at least a point of multiplicity d-r, with 1@?r@?4 and, as a consequence, every irreducible plane curve of degree d@?5 and every irreducible singular plane curve of degree 6.

Unirational fields of transcendence degree one and functional decomposition

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 <i>f</i> of the form <i>f</i> = <i>g</i>(<i>h</i>), where <i>g</i> is univariate rational function and <i>h</i> a multivariate one.

First steps towards radical parametrization of algebraic surfaces

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.

Hyperelliptic curves of genus 3 with prescribed automorphism group

We study genus 3 hyperelliptic curves which have an extra involution. The locus

On Ritt's decomposition theorem in the case of finite fields

\L_3

Covering of surfaces parametrized without projective base points

of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli

COMMON FACTORS OF RESULTANTS MODULO p

\H_3

Computation of unirational fields

. We find a birational parametrization of this locus by affine 3-space. For every moduli point

Missing sets in rational parametrizations of surfaces of revolution

\p \in \H_3