Normal forms for ordinary differential operators, II
J. Guo, A. B. Zheglov
TL;DR
This work extends the normal-form program for non-commuting ordinary differential operators by developing a Newton-region framework and applying it to derive a commutativity criterion in the Weyl algebra setting when the normal form exhibits a restriction top line. By conjugating pairs with Schur operators to reach a normal form $P'$ and using weight filtrations, the authors show that no nontrivial polynomial relation $F(P,Q)=0$ can hold in the restriction-top-line case, yielding a partial implication toward commutativity-based algebraic dependence. The results connect to Burchnall-Chaundy theory by addressing when algebraic dependence arises from commutativity and set the stage for handling the asymptotic-top-line case in future work. The work combines NR(H) geometry, a detailed combinatorial expansion for $(D+L)^k$, and a careful handling of leading terms to control commutators and algebraic relations. Collectively, it advances explicit parametrisation and commutativity criteria within the generalized Schur-normal-form framework.
Abstract
In this paper, which is a follow-up of our first paper "Normal forms for ordinary differential operators, I", we extend the theory of normal forms for non-commuting operators, and obtain as an application a commutativity criterion for operators in the Weyl algebra or, more generally, in the ring of ordinary differential operators, which we prove in the case when operators have a normal form with the restriction top line (for details see Introduction).