The highest slope of log-growth Newton polygon of p-adic differential equations

Abstract

André proved the Dwork conjecture on the log-growth Newton polygon of p-adic differential equations. We consider conditions such that the left end points of generic log-growth Newton polygon and the left end point of special log-growth Newton polygon coincide. We give an upper bound of the highest slope of differential equations at generic point. This shows that these end points coincide if differential operator has rational function coefficients and the rank of differential operator is equal to 2.