A parametric version of the Hilbert-Hurwitz theorem using hypercircles

Abstract:

Let $\mathbb {K}$ be a characteristic zero field, let $\alpha$ be an algebraic element over $\mathbb {K}$ and $\mathcal {C}$ a rational curve defined over $\mathbb {K}$ given by a parametrization $\psi$ with coefficients in $\mathbb {K}(\alpha )$. We propose an algorithm to solve the following problem, that is, a parametric version of Hilbert-Hurwitz: To compute a linear fraction $u=\frac {at+b}{ct+d}$ such that $\psi (u)$ has coefficients over an algebraic extension of $\mathbb {K}$ of degree at most two and a conic $\mathbb {K}$-birational to $\mathcal {C}$. Moreover, if the degree of $\mathcal {C}$ is odd or $\alpha$ is of odd degree over $\mathbb {K}$, we can compute a parametrization of $\mathcal {C}$ with coefficients over $\mathbb {K}$. The problem is solved without implicitization methods nor analyzing the singularities of $\mathcal {C}$.