Algebraic solutions to differential equations

When does an ordinary differential equation (ODE) have a solution which is an algebraic function, ie a function which satisfies an (non-trivial) algebraic relation with its argument? This question was asked by Fuchs in 1875, and was investigated by many mathematicians such as Schwarz and Painlevé.

In a joint work with Daniel Litt, we formulate a conjectural answer to this question, even for non-linear ODEs, in terms of arithmetic, and we prove it for a large class of ODEs at special initial conditions. I’ll try to give examples throughout to illustrate what the question has to do with arithmetic or algebraic geometry.