On the dimension of systems of algebraic difference equations

Wibmer, Michael

doi:10.1016/j.aam.2020.102136

Cited by 4 publications

(1 citation statement)

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“…One of the possible explanations why the connection between difference algebra and symbolic dynamics has widely remained unnoticed is that difference algebra has long had a strong focus on fields and integral domains. Only in the last decades the importance of studying difference algebras with many zero-divisors has been recognized, for example in [30], [20], [22], [34].…”

Section: Theorem (Difference Galois Correspondence)mentioning

confidence: 99%

Difference Galois theory and dynamics

Tomašić¹,

Wibmer²

2021

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We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference ring extensions split by a chosen Galois difference ring extension is classified by actions of the associated difference profinite Galois groupoid. In particular, difference locally étale extensions of a difference ring are classified by its difference profinite fundamental groupoid.The emergence of difference profinite spaces, viewed as discrete dynamical systems in the realm of topological dynamics, leads us to investigate the interaction of difference algebra and symbolic dynamics. As an application of this interaction, we prove the near-rationality of a certain difference zeta function counting solutions of systems of difference algebraic equations over algebraic closures of finite fields with Frobenius.

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