Descent of ordinary differential equations with rational general solutions (Q2661854)

The main focus of this paper lies on rational general solutions of algebraic ODEs defined over a differential field of rational functions \(K\). More precisely, \(K = k(t)\), where \(k\) is an algebraically closed field of characteristic zero and the derivation is the usual \( \frac{\mathrm{d}}{\mathrm{d} t} \). By a rational general solution of an algebraic ODE we understand a general solution \( \eta \) of the differential equation which is of the form \[ \eta = \frac{a_0 + a_1 t + \cdots + a_n t^n}{b_0 + b_1 t + \cdots + t^m}, \] where \(n, m\) are non-negative integers and \(a_i, b_j\) are constants in some differential extension field of \(K\). The construction of rational general solutions is of particular interest to the community of symbolic computation. Various symbolic algorithms for computing rational general solutions of algebraic ODEs have been proposed in recent years, cf.\! the introduction section of the paper and the references therein. However, there is still no algorithm for deciding the existence of a rational general solution in the general case -- even when restricting to first-order differential equations -- which makes further investigations on the structure of such solutions necessary. The present paper takes a step into this direction. In Proposition~3.2 it is shown that an algebraic ODE has a rational general solution if and only if the differential algebraic function field \( \mathcal{F} \) associated to the ODE is generated over \( K \) by constants. In this case, \( \mathcal{F} \) is actually the function field of a (non-differential) algebraic variety defined over constants (Lemma~2.2 and Remark~3.3). Therefore, the existence of rational general solutions is linked to differential descent theory, which was initiated by Matsuda and further pursued by Buium (cf.\! bibliography). Finally, the authors show that if a first-order algebraic ODE has infinitely many rational solutions (or a rational general solution), then the differential equation is of a special form (Proposition~4.4 and Corollary~4.5). In particular, any quasi-linear first-order algebraic ODE which is neither linear nor a Riccati equation cannot have a rational general solution. The latter result was already known, e.g.\! in Algorithm~2 of [\textit{N. T. Vo} et al., J. Symb. Comput. 87, 127--139 (2018; Zbl 1390.34007)] this fact constitutes a crucial step for deciding the existence of so-called strong rational general solutions. However, previous proofs relied on analytic tools such as Fuchs' theorem, whereas the derivation in this paper is purely algebraic and uses Matsuda's reformulation of movable singularities in terms of \(V\)-rings.