[For the entire collection see Zbl 0653.00005.]
The diagonal of a formal Laurent series \(F=\sum a_{n_ 1,...,n_ s}X_ 1^{n_ 1}...X_ s^{n_ s} \) over a field k is defined as \(\Delta (F)=\sum a_{n,...,n}\lambda^ n \in k((\lambda))\). A formal Laurent series \(f\in k((\lambda))\) is called diagonal of a rational function if \(f=\Delta (P/Q)\) for some polynomials P and Q. The author first reviews some known results on the algebra \({\mathcal D}(k)\subset k((\lambda))\) of diagonals of rational functions. He shows in particular that in characteristic 0 every \(f\in {\mathcal D}(k)\) is a solution of a linear differential equation with polynomial coefficients.
In the second part of the paper it is conjectured that a formal power series with integer coefficients is in \({\mathcal D}\) if and only if it has nonzero radius of convergence over \({\mathbb{C}}\) and solves a nontrivial differential equation. This conjecture is shown to be equivalent to a more geometric formulation involving differential modules over the function field of a smooth projective curve. Assuming conjectures of Y. André, Bombieri and Dwork on G-functions and related topics, the conjecture is reduced to the case of Picard-Fuchs differential equations. The main result of the paper shows that the dimension of a Picard-Fuchs type differential module M in the conjecture is at least the dimension of the r-th homology of the intersection complex of the inverse image of P; here M is defined by the Gauß-Manin connection for a \((r+1)\)- dimensional variety, projective over a smooth curve C, and a point P on C.