Systems of Discrete Differential Equations, Constructive Algebraicity of the Solutions
Résumé
In this article, we study systems of n ≥ 1, not necessarily linear, discrete differential equations (DDEs) of order k ≥ 1 with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006) who settled down the case n = 1. Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we carry out a first analysis in the direction of effectivity for solving systems of DDEs in view of practical applications.
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