Related papers: On rational double points over nonclosed fields

We show that over any field $k$, a smooth unipotent algebraic $k$-group is special if and only if it is $k$-split.

We investigate circular planar nearrings constructed from finite fields as well the complex number field using a multiplicative subgroup of order $k$, and characterize the overlaps of the basic graphs which arise in the associated…

We determine derived representation type of complete finitely generated local and two-point algebras over an algebraically closed field.

Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations $a(xb)=c$ and $a(bx)=c$ over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial…

Let $A$ be a finite-dimensional associative $k$-algebra with identity. The primary aim of this paper is to study the rationality properties of the group of all $k$-algebra automorphisms of $A$, as an affine algebraic group over an arbitrary…

The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and…

The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic $p > 5$. As a consequence, we deduce the Grauert-Riemenschneider vanishing theorem for excellent divisorial log terminal…

This paper gives a new perspective on how to solve the second-order linear differential equation written in normal form. Extending the argument of the potential to a complex number leads to solving exactly the Schr\"odinger equation when…

This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed…

In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete…

We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent…

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H)…

I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

This note presents techniques to analytically solve double integrals of the dilogarithmic type which are of great importance in the perturbative treatment of quantum field theory. In our approach divergent integrals can be calculated…

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In…

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class…

We develop explicit formulas and algorithms for arithmetic in radical function fields K/k(x) over finite constant fields. First, we classify which places of k(x) whose local integral bases have an easy monogenic form, and give explicit…

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$…