Related papers: Galois invariant smoothness basis
A new algorithms for computing discrete logarithms on elliptic curves defined over finite fields is suggested. It is based on a new method to find zeroes of summation polynomials. In binary elliptic curves one is to solve a cubic system of…
We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In…
A Lie algebroid on a variety X/k is an extension \alpha: g_X \to T_X of the tangent sheaf both as O_X-module and Lie algebra over the base field, with the obvious compatibilities; and given a Lie algebroid one has its associated ring of…
Let $k$ be a finite field, a $p$-adic field or a number field. Let $K$ be a finite extension of the Laurent series field in $m$ variables $k((x_1,...,x_m))$ or, more generally, a finite extension of the field of rational functions…
We develop an elementary method to compute spaces of equivariant maps from a homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant…
In this paper, we study Lie superalgebras of $2\times 2$ matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional…
The aim of this paper is to explain how to get a complex of smooth representations out of the dual vector space to a smooth representation of a p-adic Lie group, in natural characteristic. The construction does not depend on any…
We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as…
We consider developable surfaces along the singular set of a swallowtail which are considered to be flat approximations of the swallowtail. For the study of singularities of such developable surfaces, we introduce the notion of Darboux…
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…
The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given…
We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important…
The object of study is the group of units O^\ast(X) in the coordinate ring of a normal affine variety X over an algebraically closed field k. Methods of Galois cohomology are applied to those varieties that can be presented as a finite…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. More generally, we prove that it can be solved in the field of cardinality…
Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an O_K-order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left)…
We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the…
Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we…