Conference abstracts

Session S01 - Computational Algebra and Applications of Algebra

July 29, 17:30 ~ 17:55

## On computational aspects of the generalized differential Lüroth's theorem

### Universidad de Buenos Aires - CONICET, Argentina   -   jeronimo@dm.uba.ar

Let $\mathcal{F}$ be a differential field of characteristic $0$, $\mathbf{t} = t_1,\dots, t_m$ a set of differential indeterminates over $\mathcal{F}$, and $\mathcal{F}\langle \mathbf{t} \rangle$ the field of differential rational functions. The generalized differential Lüroth's theorem proposed by Kolchin states that, for every differential subfield $\mathcal{G}$ of $\mathcal{F}\langle \mathbf{t} \rangle$ such that the extension $\mathcal{G}/\mathcal{F}$ has differential transcendence degree $1$, there exists $v\in \mathcal{F}\langle \mathbf{t}\rangle$ with $\mathcal{G} = \mathcal{F} \langle v \rangle$. This result generalizes the differential Lüroth theorem proved by Ritt for $m=1$.

We will discuss effectivity issues of the generalized differential Lüroth theorem. If $\mathcal{G}$ is generated by a finite family of differential rational functions in $\mathcal{F}\langle \mathbf{t} \rangle$ of bounded orders and degrees, we will present upper bounds for the order and the degree of any Lüroth generator $v$ of $\mathcal{G}$ over $\mathcal{F}$. These are the first known bounds for arbitrary $m$ and, in the case $m=1$, they improve the previous degree bounds. In addition, we will show that a Lüroth generator can be computed by means of classical techniques from computer algebra applied to a polynomial ideal associated with the given generators. Finally, we will show how to determine whether a given differentially finitely generated subfield of $\mathcal{F}\langle \mathbf{t} \rangle$ has differential transcendence degree $1$ over $\mathcal{F}$.

Joint work with Lisi D'Alfonso (Universidad de Buenos Aires, Argentina) and Pablo Solernó (Universidad de Buenos Aires - CONICET, Argentina).