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Mathematics of Computation

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Estimates for the coefficients of differential dimension polynomials

Author: Omar León Sánchez
Journal: Math. Comp. 88 (2019), 2959-2985
MSC (2010): Primary 12H05, 14Q20
Published electronically: March 28, 2019
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Abstract: We answer the following longstanding question of Kolchin: given a system of algebraic-differential equations $ \Sigma (x_1,\dots ,x_n)=0$ in $ m$ derivatives over a differential field of characteristic zero, is there a computable bound that only depends on the order of the system (and on the fixed data $ m$ and $ n$) for the typical differential dimension of any prime component of $ \Sigma $? We give a positive answer in a strong form; that is, we compute a (lower and upper) bound for all the coefficients of the Kolchin polynomial of every such prime component. We then show that, if we look at those components of a specified differential type, we can compute a significantly better bound for the typical differential dimension. This latter improvement comes from new combinatorial results on characteristic sets, in combination with the classical theorems of Macaulay and Gotzmann on the growth of Hilbert-Samuel functions.

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Additional Information

Omar León Sánchez
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

Keywords: Kolchin polynomial, typical differential dimension, Hilbert-Samuel functions
Received by editor(s): March 1, 2017
Received by editor(s) in revised form: August 10, 2017, April 27, 2018, and January 24, 2019
Published electronically: March 28, 2019
Article copyright: © Copyright 2019 American Mathematical Society