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.
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We consider a differential system xd dxY=GY, where G is a m×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=Fexp(G(0)logx) 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.

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DOI : 10.4171/rsmup/95
Classification : 12 
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     author = {Nakagawa, Takahiro},
     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://archive.numdam.org/articles/10.4171/rsmup/95/}
}
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Nakagawa, Takahiro. 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://archive.numdam.org/articles/10.4171/rsmup/95/

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