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Algorithms yield upper bounds in differential algebra

Part of: Model theory Computability and recursion theory Differential and difference algebra

Published online by Cambridge University Press:  29 September 2021

Wei Li ,
Alexey Ovchinnikov ,
Gleb Pogudin  and
Thomas Scanlon
Wei Li
Affiliation:
KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Beijing 100190, China e-mail: liwei@mmrc.iss.ac.cn
Alexey Ovchinnikov*
Affiliation:
Department of Mathematics, CUNY Queens College, 65-30 Kissena Boulevard, Queens, NY 11367, USA Ph.D. Programs in Mathematics and Computer Science, CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA
Gleb Pogudin
Affiliation:
Institut Polytechnique de Paris, LIX, CNRS, École Polytechnique, 1 rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia e-mail: gleb.pogudin@polytechnique.edu
Thomas Scanlon
Affiliation:
Department of Mathematics, University of California—Berkeley, Berkeley, CA 94720-3840, USA email: scanlon@math.berkeley.edu
*
e-mail: aovchinnikov@qc.cuny.edu
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Abstract

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including, for example, difference fields).

We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

Keywords

delay PDEselimination of unknownsuniform boundsoracle machine

MSC classification

Primary: 12H05: Differential algebra 12H10: Difference algebra 03C10: Quantifier elimination, model completeness and related topics
Secondary: 03C60: Model-theoretic algebra 03D15: Complexity of computation (including implicit computational complexity)
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 29 - 51
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work was partially supported by the NSF grants CCF-1564132, CCF-1563942, DMS-1760448, DMS-1760413, DMS-1853650, DMS-1853482, and DMS-1800492; the NSFC grants 11971029, 12122118 and 11688101; and the fund of Youth Innovation Promotion Association of CAS.

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