On Abel's Problem about Logarithmic Integrals in Positive Characteristic

Abstract

Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension ${\mathcal{R}}_p=K(z_1,z_2,\dots )((x))$ of $K(x)$, where $x^{\prime }=1,z_1^{\prime }=1/x$ and $z_i^{\prime }=z_{i-1}^{\prime }/z_{i-1}$. For differential equations of order $1$ it is shown that there exists a solution $y$ whose projections $y{|}_{z_{i+1}=z_{i+2}=\cdots =0}$ are algebraic over the field of rational functions $K(x,z_1,\dots ,z_i)$ for all $i$. This can be seen as a characteristic $p$ analogue of Abel's problem about the algebraicity of logarithmic integrals. Further, the existence of infinite product representations of these solutions is shown. As a main tool $p^i$-curvatures are introduced, generalizing the notion of the $p$-curvature.

28 pages