Elimination of unknowns for systems of algebraic differential-difference equations
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- by Wei Li, Alexey Ovchinnikov, Gleb Pogudin and Thomas Scanlon PDF
- Trans. Amer. Math. Soc. 374 (2021), 303-326 Request permission
Abstract:
We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function $B(r,s)$ of the natural number parameters $r$ and $s$ so that for any system of algebraic ordinary differential-difference equations in the variables $\mathbfit {x} = x_1, \ldots , x_q$ and $\mathbfit {y} = y_1, \ldots , y_r$, each of which has order and degree in $\mathbfit {y}$ bounded by $s$ over a differential-difference field, there is a nontrivial consequence of this system involving just the $\mathbfit {x}$ variables if and only if such a consequence may be constructed algebraically by applying no more than $B(r,s)$ iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over $\mathbb {C}$ is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.References
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Additional Information
- Wei Li
- Affiliation: KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55 Zhongguancun East Road, 100190, Beijing, People’s Republic of China
- Email: liwei@mmrc.iss.ac.cn
- Alexey Ovchinnikov
- Affiliation: Department of Mathematics, CUNY Queens College, 65-30 Kissena Boulevard, Queens, New York 11367; CUNY Graduate Center, Mathematics and Computer Science, 365 Fifth Avenue, New York, New York 10016
- MR Author ID: 736046
- ORCID: 0000-0001-8192-910X
- Email: aovchinnikov@qc.cuny.edu
- Gleb Pogudin
- Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
- Address at time of publication: LIX, CNRS, École Polytechnique, Institut Polytechnique de Paris, France
- MR Author ID: 948033
- Email: gleb.pogudin@polytechnique.edu
- Thomas Scanlon
- Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
- MR Author ID: 626736
- ORCID: 0000-0003-2501-679X
- Email: scanlon@math.berkeley.edu
- Received by editor(s): December 29, 2018
- Received by editor(s) in revised form: March 18, 2020
- Published electronically: October 14, 2020
- Additional Notes: This work has been partially supported by NSF under grants CCF-1564132, CCF-1563942, DMS-1760448, DMS-1760413, DMS-1800492, DMS-1853482, and DMS-1853650; PSC-CUNY under grant 60098-00 48; and NSFC under grants 11971029, 11688101.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 303-326
- MSC (2010): Primary 12H05, 12H10, 03C10; Secondary 14Q20
- DOI: https://doi.org/10.1090/tran/8219
- MathSciNet review: 4188184