Field Of Zero Research Articles

Module structure of the homology of right-angled Artin kernels

In this paper, we study the module structure of the homology of Artin kernels, i.e., kernels of non-resonant characters from right-angled Artin groups onto the integer numbers, the module structure being with respect to the ring $\mathbb{K}[t^{\pm 1}]$, where $\mathbb{K}$ is a field of characteristic zero. Papadima and Suciu determined some part of this structure by means of the flag complex of the graph of the Artin group. In this work, we provide more properties of the torsion part of this module, e.g., the dimension of each primary part and the maximal size of Jordan forms (if we interpret the torsion structure in terms of a linear map). These properties are stated in terms of homology properties of suitable filtrations of the flag complex and suitable double covers of an associated toric complex.

A Local Trace Formula for p-adic Infinitesimal Symmetric Spaces: the Case of Guo–Jacquet

Abstract We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo–Jacquet trace formulae, and our methods are inspired by works of Waldspurger and Arthur. Some other results are given during the proof including a noninvariant local trace formula, Howe’s finiteness for weighted orbital integrals, and the representability of the Fourier transform of weighted orbital integrals. These local results are preparations for the comparison of regular semi-simple terms, which are weighted orbital integrals, appearing in an infinitesimal variant of Guo–Jacquet trace formulae.

Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

Let $\Sigma$ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-$\Sigma$ fundamental group of the configuration space. Let $X$ be a hyperbolic curve of type $(g,r)$ over a field $k$ of characteristic zero. Thus, $X$ is obtained by removing from a proper smooth curve of genus $g$ over $k$ a closed subscheme [i.e., the "divisor of cusps"] of $X$ whose structure morphism to Spec$(k)$ is finite étale of degree $r$; $2g-2+r>0$. Write $X_n$ for the $n$-th configuration space associated to $X$, i.e., the complement of the various diagonal divisors in the fiber product over $k$ of $n$ copies of $X$. Then, when $k$ is algebraically closed, we show that the triple $(n,g,r)$ and the generalized fiber subgroups—i.e., the subgroups that arise from the various natural morphisms $X_n \to X_m$ $[m$ < $n]$, which we refer to as generalized projection morphisms—of the pro-$\Sigma$ fundamental group $\Pi_n$ of $X_n$ may be reconstructed group-theoretically from $\Pi_n$ whenever $n \geq 2$. This result generalizes results obtained previously by the first and third authors and A. Tamagawa to the case of arbitrary hyperbolic curves [i.e., without restrictions on $(g,r)$]. As an application, in the case where $(g,r)= (0,3)$ and $n \geq 2$, we conclude that there exists a direct product decomposition Out$(\Pi_n) = {\mathrm{GT}^{\Sigma}} \times \mathfrak{S}_{n + 3}$ —where we write "Out$(-)$" for the group of outer automorphisms [i.e., without any auxiliary restrictions!] of the profinite group in parentheses and ${\mathrm{GT}^{\Sigma}}$ (respectively, $\mathfrak{S}_{n + 3}$) for the pro-$\Sigma$ Grothendieck-Teichmüller group (respectively, symmetric group on $n + 3$ letters). This direct product decomposition may be applied to obtain a simplified purely group-theoretic equivalent definition—i.e., as the centralizer in ${\rm Out}(\Pi_n)$ of the union of the centers of the open subgroups of ${\rm Out}(\Pi_n)$—of ${\mathrm{GT}^{\Sigma}}$. One of the key notions underlying the theory of the present paper is the notion of a pro-$\Sigma$ log-full subgroup—which may be regarded as a sort of higher-dimensional analogue of the notion of a pro-$\Sigma$ cuspidal inertia subgroup of a surface group—of $\Pi_n$. In the final section of the present paper, we show that, when $X$ and $k$ satisfy certain conditions concerning "weights", the pro-$l$ log-full subgroups may be reconstructed group-theoretically from the natural outer action of the absolute Galois group of $k$ on the geometric pro-$l$ fundamental group of $X_n$.

Nonbirational centers of linear projections of scrolls over curves

We work over an algebraically closed field of characteristic zero. A nonbirational center of a projective variety is a point from which the variety is projected nonbirationally onto its image, whose locus plays an important role in a study of the double-point divisors and the defining equations of the variety. The purpose of this paper is to show that a scroll over a curve with some conditions has no nonbirational centers. Consequently such a nondegenerate scroll is cutting out by hypersurfaces of degree $d-e + 1$ for its degree $d$ and codimension $e$ in the projective space. On the other hand, examples of scrolls over curves with nonbirational centers are constructed.

Classification of naturally graded nilpotent associative algebras

In the paper, we consider dimensional naturally graded nilpotent associative algebras with the characteristic sequence over an algebraically closed field of characteristic zero. There are two types of such algebras. We give the algebraic classification up to isomorphism of both types of algebras. Moreover, we prove that there are no such algebras of the second type for .

Seshadri constants over fields of characteristic zero

Let X be a smooth projective variety defined over a field k of characteristic 0 and let L be a nef line bundle defined over k. We prove that if x∈X is a k-rational point then the Seshadri constant ε(X,L,x) over k‾ is the same as that over k. We show, by constructing families of examples, that there are varieties whose global Seshadri constant ε(X) is zero. We also prove a result on the existence of a Seshadri curve with a natural (and necessary) hypothesis.

Visualization and Analysis of Global Vision Zero Studies and Policy Orientation in China.

As a policy that is widely used worldwide, Vision Zero is a long-term strategic goal for road safety in China. The aim of this paper is to examine the literature in the field of Vision Zero retrieved by the Web of Science (WOS) Core Collection database from 1997 to 2021 based on mapping knowledge domain (MKD) and bibliometric methods. In this paper, we analyzed the overall development level of Vision Zero at different stages using the statistical analysis of the distribution of literature years. Based on the analysis, four major research directions of Vision Zero are obtained through keywords co-occurrence analysis, including "Vision Zero in road safety", "vulnerable road users' risks", "effect of speed on severity", and "responsibility management for crashes". Furthermore, we explore the influence and development potential of each country and region of Vision Zero based on the number and citation frequency of the literature, and the distribution of time dimensions. Among the research topics, all countries and regions are clustered into four clusters, and the current status of Vision Zero for countries in the most influential clusters, which include Sweden, the United States, Australia, China, and Norway, have been presented. Finally, an analysis of road safety in China is carried out, which includes displaying the changes in the number of road accidents, fatalities, and serious injuries in China between 2009 and 2018, comparing the fatalities per 10,000 vehicles and the fatalities per 100,000 people between China and other countries, and describing China's strategic routes toward Vision Zero.

Zariski’s dimensionality type of singularities. Case of dimensionality type 2

In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive “generic” corank 1 1 projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In this paper we completely settle the case of dimensionality type 2, by studying Zariski equisingular families of surfaces singularities, not necessarily isolated, in the three-dimensional space.

Equivariant Cobordism of Smooth Projective Spherical Varieties

AbstractWe study the equivariant cobordism rings for the action of a torusTon smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rationalT-equivariant cobordism rings of smooth projectiveG-spherical varieties with the action of a maximal torusTofG. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.

Images of graded polynomials on matrix algebras

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra Mn(K) over a field K endowed with its canonical Zn-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field Q of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on n×n matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over Mn(K) when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on M2(K).

On existence of PI-exponent of algebras with involution

We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of ⁎-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional ⁎-PI-exponent. We also construct a family of infinite-dimensional algebras Cα such that exp⁎(Cα) does not exist.

Linear periods for unitary representations

Let F be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadić’s classification of the unitary dual of \({\mathrm {GL}}_{2n}(F)\), we classify irreducible unitary representations of \({\mathrm {GL}}_{2n}(F)\) that have nonzero linear periods, in terms of Speh representations that have nonzero periods. We also give a necessary and sufficient condition for the existence of a nonzero linear period for a Speh representation.

An effective Schmidt’s subspace theorem for arbitrary hypersurfaces over function fields

We establish an effective version of Schmidt’s subspace theorem on a smooth projective variety [Formula: see text] over function fields of characteristic zero for arbitrary family of hypersurfaces. We also show an application to study a general Thue equation over function fields.

Graded monomial identities and almost non-degenerate gradings on matrices

Let F be a field of characteristic zero, G be a group and Mn(F) be the algebra of matrices of size n with entries from F with a G-grading. Bahturin and Drensky proved that if the G grading on Mn(F) is elementary and the neutral component of Mn(F) is commutative, then the graded identities of Mn(F) follow from three basic types of identities and monomial identities of length ≥2 bounded by a function f(n) of n. In this paper we prove the best upper bound is f(n)=n. More generally, we prove that all the graded monomial identities of an elementary G-grading on Mn(F) follow from those of degree at most n. We also study gradings which satisfy no graded multilinear monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on Mn(F) is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate Z-gradings on Mn(F) for n≤5.

On initials and the fundamental theorem of tropical partial differential algebraic geometry

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions.We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial.

The structure of connected (graded) Hopf algebras revisited

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K-module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].

On representations induced from the Zelevinsky segment and a tempered representation in the half-integral case

Let [Formula: see text] denote either the group [Formula: see text] or [Formula: see text] over a non-archimedean local field of characteristic zero. We determine the reducibility criteria for a parabolically induced representation of the form [Formula: see text], where [Formula: see text] denotes the Zelevinsky segment representation of the general linear group attached to the segment [Formula: see text], with [Formula: see text] half-integral, and [Formula: see text] denotes an irreducible tempered representation of [Formula: see text].

POINTWISE INNER AUTOMORPHISMS OF RELATIVELY FREE LIE ALGEBRAS

Let $K$ be a field of characteristic zero, and $L_{m,c}$ be the free metabelian nilpotent Lie algebra of class $c$ of rank $m$ over $K$. We call an automorphism $\phi$ pointwise inner, if there exists an inner automorphism $\xi_i$ for each generator $x_i$, $i=1,\ldots,m$, such that $\phi(x_i)=\xi_i(x_i)$. In this study, we exemine the group $PI(L_{m,c})$ of pintwise inner automorphisms of the Lie algebra $L_{m,c}$, and we provide a set of generators for this group.

Sum of two strictly n-zero matrices

In this work we investigate when an Jordan matrix can be written as the sum of two strictly n-zero matrices. In particular, we show that if is an algebraically closed field of characteristic zero and A is an matrix over with , then A is the sum of two strictly n-zero matrices.

Identities and cocharacters of the algebra of 3 × 3 matrices with non-trivial grading and transpose graded involution

Let F be a field of characteristic zero and M2,1(F) the algebra of 3×3 matrices over F endowed with non-trivial Z2-grading. The transpose involution t on M2,1(F) preserves the homogeneous components of the grading and so, we consider (M2,1(F),t) as a superalgebra with graded involution. We study the (Z2,⁎)-identities of this algebra and make explicit the decomposition of the space of multilinear (Z2,⁎)-identities into the sum of irreducibles under the action of the group (Z2×Z2)≀Sn in order to determine all the irreducible characters appearing with non-zero multiplicity in the decomposition of the ⁎-graded cocharacter of (M2,1(F),t). Along the way, using the representation theory of the general linear group, we determine all the (Z2,⁎)-identities of (M2,1(F),t) up to degree 3.