Normal forms for ordinary differential operators, I

2024/06/20 by Guo, J., Zheglov, A. B. · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2406.14414

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 D₁=K[[x]][∂ ] as a subring of a certain complete non-commutative ring D₁^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.

Cited by

• Normal forms for ordinary differential operators, II
• Normal forms for ordinary differential operators, III

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