2024/01/25 by Florian Fürnsinn, Fürnsinn, Florian, H. Hauser +3
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.14154
openalex publication_date 2024/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Rp=K(z1, z2, …)( ( x) ) of K(x), where x'=1, z1'=1/x and zi'=zi-1'/zi-1. For differential equations of order 1 it is shown that there exists a solution y whose projections y\vert_zi+1=zi+2=⋯=0 are algebraic over the field of rational functions K(x, z1, …, zi) 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 pi-curvatures are introduced, generalizing the notion of the p-curvature.