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SOLVING DIFFERENCE EQUATIONS IN SEQUENCES: UNIVERSALITY AND UNDECIDABILITY

Part of: Computational aspects in algebraic geometry Difference and functional equations Computability and recursion theory Computational aspects and applications Differential and difference algebra Difference equations

Published online by Cambridge University Press:  30 June 2020

GLEB POGUDIN ,
THOMAS SCANLON  and
MICHAEL WIBMER
GLEB POGUDIN
Affiliation:
Department of Computer Science, National Research University Higher School of Economics, Moscow, Russia; [email protected]
THOMAS SCANLON
Affiliation:
University of California at Berkeley, Department of Mathematics, Berkeley, USA; [email protected]
MICHAEL WIBMER
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria; [email protected]

Abstract

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We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

MSC classification

Primary: 12H10: Difference algebra
Secondary: 39A10: Difference equations, additive 13P25: Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) 14Q20: Effectivity, complexity 03D35: Undecidability and degrees of sets of sentences
Type
Differential Equations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e33
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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