Comparison between formal slopes and p-adic slopes

In this paper, we establish several inequalities comparing formal slopes with p-adic slopes of solvable differential modules over the punctured open unit disc. Our approach is based on a delicate analysis of Newton polygons and the log-convexity of generic radius functions.

💡 Research Summary

This paper investigates the relationship between two invariants attached to solvable differential modules over the punctured open unit disc: formal slopes (coming from the formal classification over the Laurent series field K = k((x))) and p‑adic slopes (arising from the p‑adic analytic theory over the Robba ring). Let M be a solvable differential module of rank n over the subring Aₓ of the Robba ring, which embeds naturally both into K and into the Robba ring R. Consequently, both formal slopes β₁ ≥ … ≥ βₙ and p‑adic slopes α₁ ≥ … ≥ αₙ are defined for M. The main result (Theorem 1.1) asserts that for every i = 1,…,n the partial sums satisfy

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