Related papers: Riemann-Roch for the ring $\mathbb Z$

We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to…

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of $\mathbb S[\mu_{n,+}]$-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring $\mathbb Z$…

We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The…

Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting…

Some basic properties of the ring of integers $\mathbb{Z}$ are extended to entire rings. In particular, arithmetic in entire principal rings is very similar than arithmetic in the ring of integers $\mathbb{Z}$. These arithmetic properties…

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…

Motivated by a valuation theorem, recently obtained by Rangachev, we study the \'etale extensions $A\subset B$ of polynomial rings over an algebraically closed field of characteristic zero, such that the integral closure $\overline{A}$ is a…

We prove an equivariant Riemann-Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in Q. We…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is…

We show that every polynomial overring of the ring ${\rm Int}(\mathbb Z)$ of polynomials which are integer-valued over $\mathbb Z$ may be considered as the ring of polynomials which are integer-valued over some subset of $\hat{\mathbb{Z}}$,…

We study the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. The ring of polyfunctions over a commutative ring $R$ with unit element is the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense…

Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in…

It is often stated that the Carlitz module is to the ring of univariate polynomials over a finite field what the multiplicative group is to the ring of integers. This analogy extends to the "rank 2" case, where Drinfeld modules play a role…

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…

In this paper we study the arithmetic invariants of Euclidean lattice in the context of Arakelov geometry. We regard a Euclidean lattice as a hermitian vector bundle $\bar E$ on ${\rm Spec}(\mathbb{Z})$ and consider two typical arithmetic…

Consider the ring $\mathcal{S}$ of symmetric polynomials in $k$ variables over an arbitrary base ring $\mathbf{k}$. Fix $k$ scalars $a_{1},a_{2},\ldots,a_{k}\in\mathbf{k}$. Let $I$ be the ideal of $\mathcal{S}$ generated by…

A representation of SL(2,Z) by integer matrices acting on the space of analytic ordinary Dirichlet series is constructed, in which the standard unipotent element acts as multiplication by the Riemann zeta function. It is then shown that the…

The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N…

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\ldots,x^{p_t}y^{s_t},y^b] \subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion…