Related papers: Grothendieck rings of \mathbb{Z}-valued fields

We study Grothendieck rings (in the sense of logic) of fields. We prove the triviality of the Grothendieck rings of certain fields by constructing definable bijections which imply the triviality. More precisely, we consider valued fields,…

We consider the model-theoretic Grothendieck ring of definable sets in ordered abelian groups. It is well-known that $\mathrm{K} \mathbb{Q} \cong \mathbb{Z}[T]/(T^2 + T)$ and $\mathrm{K} \mathbb{Z} =0$, but surprisingly little is known…

We construct, for every integer $N\in\mathbb{N}^*$, a structure whose Grothendieck ring is isomorphic to $(\mathbb{Z}/N\mathbb{Z})[X]$, thus proving the existence of structures with a non-zero Grothendieck ring with non-zero characteristic.…

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction…

Let R be a regular local ring, containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R.

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. If the regular local ring R contains an…

In this article, we establish the Grothendieck-Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This…

In this paper, we present a captivating construction by Grothendieck, originally formulated for algebraic varieties, and adapt it to the realm of C*-algebras. Our main objective is to investigate the conditions under which this particular…

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the…

Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by…

Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K^m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and…

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R…

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category $\mathcal{O}$ of representations of the quantum loop algebra introduced by Hernandez-Jimbo. We use the cluster algebra structure of the…

The Weyl algebra over a field $k$ of characteristic $0$ is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all $\mathbb{Z}$-graded simple rings of GK-dimension 2 and show that they…

This note contains another proof of Grothendieck`s theorem on the splitting of vector bundles on the projective line over a field $k$. Actually the proof is formulated entirely in the classical terms of a lattice $\Lambda \cong k[T]^d$,…

We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of…

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the…

Assume that R is a semi-local regular ring containing an infinite perfect field, or that R is a semi-local ring of several points on a smooth scheme over an infinite field. Let K be the field of fractions of R. Let H be a strongly inner…