Abstract
We study the problem of formal decomposition (non-commutative factorization) of linear ordinary differential operators over the field \({{\mathbb {C}}}(\!(t)\!)\) of formal Laurent series at an irregular singular point corresponding to \(t=0\). 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.
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Notes
The classical Weyl algebra is generated by two symbols with the same commutation relation, so consists of noncommutative polynomials in these variables.
The classical notion of the symbol of a differential operator collects only the terms involving the highest order derivatives.
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Acknowledgements
This paper is a product of a thorough rethinking of the thesis of the first author Mezuman (2017). We are grateful to many friends and colleagues who came out with most helpful remarks after hearing conference presentations of the results, especially Jean-Pierre Ramis, Michael Singer, Daniel Bertrand, Gal Binyamini and Dmitry Novikov. The anonymous referee did a great job spotting even the smallest ambiguities and typos: we are most sincerely grateful to her/him. The second author is incumbent of the Gershon Kekst Chair of Mathematics.
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To Askold Khovanskii, a lifelong role model and close friend, for his 70th birthday.
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Mezuman, L., Yakovenko, S. Formal Factorization of Higher Order Irregular Linear Differential Operators. Arnold Math J. 4, 279–299 (2018). https://doi.org/10.1007/s40598-019-00104-z
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DOI: https://doi.org/10.1007/s40598-019-00104-z