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Formal Factorization of Higher Order Irregular Linear Differential Operators

Formal Factorization of Higher Order Irregular Linear Differential Operators We study the problem of formal decomposition (non-commutative factorization) of linear ordinary differential operators over the field $${{\mathbb {C}}}(\!(t)\!)$$ C ( ( t ) ) of formal Laurent series at an irregular singular point corresponding to $$t=0$$ t = 0 . The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an “automatic translation” which allows to obtain the results for formal factorization in the Weyl algebra from well known results in local analytic geometry. In addition, we draw some (apparently unnoticed) parallels between the formal factorization of linear operators and formal diagonalization of systems of linear first order differential equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

Formal Factorization of Higher Order Irregular Linear Differential Operators

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References (6)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Institute for Mathematical Sciences (IMS), Stony Brook University, NY
Subject
Mathematics; Mathematics, general
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-019-00104-z
Publisher site
See Article on Publisher Site

Abstract

We study the problem of formal decomposition (non-commutative factorization) of linear ordinary differential operators over the field $${{\mathbb {C}}}(\!(t)\!)$$ C ( ( t ) ) of formal Laurent series at an irregular singular point corresponding to $$t=0$$ t = 0 . The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an “automatic translation” which allows to obtain the results for formal factorization in the Weyl algebra from well known results in local analytic geometry. In addition, we draw some (apparently unnoticed) parallels between the formal factorization of linear operators and formal diagonalization of systems of linear first order differential equations.

Journal

Arnold Mathematical JournalSpringer Journals

Published: Mar 18, 2019

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