Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation 2016
DOI: 10.1145/2930889.2930912
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On p-Adic Differential Equations with Separation of Variables

Abstract: Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate p-adic setting to be well-posed. This raises precision concerns: how much precision do we need on the input to compute the output accurately? In the case of ordinary differential equations with separation of variables, we make use of the recent technique of differential precision to obtain optimal bounds on the sta… Show more

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Cited by 14 publications
(31 citation statements)
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“…A complete solution is described in [15]. More generally, a framework for using p-adic arithmetic to solve ordinary differential equations in positive characteristic may be found in [32].…”
Section: Previous Workmentioning
confidence: 99%
“…A complete solution is described in [15]. More generally, a framework for using p-adic arithmetic to solve ordinary differential equations in positive characteristic may be found in [32].…”
Section: Previous Workmentioning
confidence: 99%
“…In [10], the behavior of the precision when solving p-adic di erential equations with separation of variables has been studied. The authors have investigated the gap that appears when applying a Newton-method solver between the theoretic loss in precision and the actual loss in precision for a naive implementation in Zp(p).…”
Section: P-adic DI Erential Equationsmentioning
confidence: 99%
“…(We then encourage the reader to read it and study it again in light of the discussion of this paragraph.) More recently Lairez and Vaccon applied this strategy for the computation of the solutions of p-adic differential equations with separation of variables [53]; they managed to improve this way a former algorithm by Lercier and Sirvent for computing isogenies between elliptic curves in positive characteristic [57].…”
Section: Corollary 335mentioning
confidence: 99%
“…The method of adaptive precision (presented in §3.3.1) has been used with success in several concrete contexts: the computation of the solutions of some p-adic differential equations [53] and the computation of GCD's and Bézout coefficients via the usual extended Euclidean algorithm [17]. Below, we detail another example, initially pointed out by Buhler and Kedlaya [15], which is simpler (it actually looks like a toy example) but already shows up all the interesting phenomena: it is the evaluation of the p-adic Somos 4 sequences.…”
Section: Example: the Somos 4 Sequencementioning
confidence: 99%