A topos-theoretic view of difference algebra

Tomašić, Ivan

doi:10.48550/arxiv.2001.09075

Cited by 3 publications

(4 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nevertheless, we find it useful to view it as an undercategory/coslice of σ-Rng under k, k-Alg ≃ k/σ-Rng. for the underlying small category with objects ⌊C 0 ⌋ and morphisms ⌊C 1 ⌋, the morphisms σ C0 and σ C1 define a functor σ C : C → C , so a difference category C can be identified with the datum (C , σ C ), as noted in [28]. Definition 2.12.…”

Section: 1mentioning

confidence: 99%

“…On the other hand, the first author developed an approach to difference algebra through topos theory and categorical logic in [28], and the key idea from those areas is that in contexts with not enough points ('fixed points' when it comes to difference sets), everything is governed by internal localic groupoids, as in [14], [5], [3], [4]. In a forthcoming paper, the first author will formulate the theory of the difference localic fundamental group reliant on the general theory from [4].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Difference Galois theory and dynamics

Tomašić¹,

Wibmer²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

Difference Galois theory and dynamics

Tomašić¹,

Wibmer²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the algebraic theory of difference equations there has long been a focus on fields, but in the last decade the importance of studying solutions of systems of algebraic difference equations in more general difference rings has more and more been recognized. See e.g., [vdPS97,Hru04,Tom14,Tom16,MS11,DVHW14,Wibb,Tom]. In particular, the solution sets of systems of algebraic difference equations in the ring of sequences, which are of utmost importance from an applied perspective, have been studied in [OPS20] and [PSW20].…”

Section: Introductionmentioning

confidence: 99%

On the dimension of systems of algebraic difference equations

Wibmer

2020

Preprint

View full text Add to dashboard Cite

We introduce a notion of dimension for the solution set of a system of algebraic difference equations that measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but, as we show, it satisfies properties suitable for a notion of dimension. We also show that the dimension of a difference monomial is given by the covering density of its set of exponents.

show abstract

“…After finishing this paper, it was pointed out to us by Ivan Tomašić that our left difference sheaves correspond to certain objects which were identified by him in topos-theoretic terms (see [25,Section 22.1]). It remains to be seen whether our right difference sheaves have a similar interpretation.…”

mentioning

confidence: 99%