On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents

Takahiro Nakagawa

doi:10.4171/rsmup/95

On log-growth of solutions of

p

-adic differential equations with

p

-adic exponents

Takahiro Nakagawa

Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 153-168.

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Résumé

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

Accepté le : 2020-07-02
Publié le : 2022-07-11

MR Zbl

DOI : 10.4171/rsmup/95

Classification : 12

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     author = {Takahiro Nakagawa},
     title = {On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {153--168},
     volume = {147},
     year = {2022},
     doi = {10.4171/rsmup/95},
     mrnumber = {4450788},
     zbl = {1495.12003},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/95/}
}

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Takahiro Nakagawa. On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents. Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 153-168. doi : 10.4171/rsmup/95. http://www.numdam.org/articles/10.4171/rsmup/95/

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