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arXiv · 2406.14414

Normal forms for ordinary differential operators, I

J. Guo·A. B. Zheglov
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Abstract

In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups. This parametrisation is obtained with the help of normal forms - a notion we introduce in this paper. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete non-commutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for any given commuting operators.

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J. Guo, A. B. Zheglov. 2025-11-04. Normal forms for ordinary differential operators, I. https://arxiv.org/abs/2406.14414

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