On a Problem of Polya and Some of its Evolutions

Around 1921 Polya asked about the algebraicity of the indefinite integral of an algebraic function, in terms of the integrality of the coefficients in a power-series expansion. Polya nicely answered the special case of rational functions, using Fermat's congruence. Though the formulation was very simple, the general case was solved only much later by transcendence techniques.  The question is also linked with a simply stated well-known local-global conjecture of Grothendieck on linear differential equations over polynomial rings, and with other number-theoretical issues.

We shall survey on the topic, also illustrating a novel question arising from a different possible approach to Polya's issue, and some answers given more recently by Katz. If time allows we shall sketch a recent argument to prove a strengthening of the main statements.