JournalsrsmupVol. 147pp. 153–168

On log-growth of solutions of pp-adic differential equations with pp-adic exponents

  • Takahiro Nakagawa

    Chiba University, Japan
On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents cover
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Abstract

We consider a differential system xddxY=GYx\frac{d}{dx} Y=GY, where GG is a m×mm\times m matrix whose coefficients are power series which converge and are bounded on the open unit disc D(0,1)D(0,1^-). Assume that G(0)G(0) is a diagonal matrix with pp-adic integer coefficients. Then there exists a solution matrix of the form Y=Fexp(G(0)logx)Y=F \exp(G(0)\log x) at x=0x=0 if all differences of exponents of the system are pp-adically non-Liouville numbers. We give an example where FF is analytic on the pp-adic open unit disc and has log-growth greater than mm. Under some conditions, we prove that if a solution matrix at a generic point has log-growth δ\delta, then FF has log-growth δ\delta.

Cite this article

Takahiro Nakagawa, On log-growth of solutions of pp-adic differential equations with pp-adic exponents. Rend. Sem. Mat. Univ. Padova 147 (2022), pp. 153–168

DOI 10.4171/RSMUP/95