Spectral Curves for Third-Order ODOs
Abstract
:1. Introduction
2. Contributions
- The principal ideal generated by f in equals the prime ideal .
- The differential ideal generated by f in equals the radical of the elimination ideal .
3. Algebro-Geometric ODOs
- L has a nontrivial centralizer .
- There exists an operator A in of order m, relatively prime with 3, such that .
4. Burchnall–Chaundy Ideals and Spectral Curves
4.1. Burchnall–Chaundy Ideal of a Pair
- is a commutative domain isomorphic to , the coordinate ring of the spectral curve .
- There exists an irreducible polynomial such that .
4.2. Burchnall–Chaundy Ideal of a Third-Order Operator
5. Elimination Ideals for Commuting Pairs of ODOs
5.1. Generalized Previato–Wilson Theorem
5.2. Computing the Burchnall–Chaundy Ideal of a Pair
- The radical of the elimination ideal equals .
- The radical of the elimination ideal equals .
6. Centralizers as Coordinate Rings
6.1. Normalized Basis
6.2. Generators of the Burchnall–Chaundy Ideal of a Third-Order ODO
- Irreducible. If is irreducible, then and is an irreducible curve.
- Non-irreducible. If is not irreducible, then properly divides r and . Including in the ideal allows us to select one irreducible component of . The example in Section 8 illustrates this situation.
- The polynomial is an implicit representation of the Zariski closure of the projection of γ onto the plane . Furthermore, because is irreducible over . Thus, this projection is an irreducible algebraic curve .
- On the other hand, since this projection is assumed birational on γ, then contains a linear polynomial in λ, say .
7. Parametric Factorization of Algebro-Geometric ODOs
7.1. Coefficient Field for Factorization
7.2. The Intrinsic Right Factor
8. Example of Non-Planar Spectral Curve
9. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Centralizers of Ordinary Differential Operators
- There exists a nonzero constant such that .
- We have , and .
- If , there exists a nonzero constant such that .
- the rank of as a -module divides ;
- is a (commutative) differential domain.
Appendix B. Differential Resultant of Two ODOs
- .
- .
- P and Q are right coprime in .
Appendix B.1. First Differential Subresultant
- is a differential operator of order 1;
- equals up to multiplication by an element of .
Appendix B.2. Characterizing Common Solutions
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Rueda, S.L.; Zurro, M.-A. Spectral Curves for Third-Order ODOs. Axioms 2024, 13, 274. https://doi.org/10.3390/axioms13040274
Rueda SL, Zurro M-A. Spectral Curves for Third-Order ODOs. Axioms. 2024; 13(4):274. https://doi.org/10.3390/axioms13040274
Chicago/Turabian StyleRueda, Sonia L., and Maria-Angeles Zurro. 2024. "Spectral Curves for Third-Order ODOs" Axioms 13, no. 4: 274. https://doi.org/10.3390/axioms13040274
APA StyleRueda, S. L., & Zurro, M. -A. (2024). Spectral Curves for Third-Order ODOs. Axioms, 13(4), 274. https://doi.org/10.3390/axioms13040274