Even points on an algebraic curve

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.     

Simultaneous approximation to values of exponential function at algebraic points

This paper applies the methods of Chudnovsky^[2], Osgood^[4] and Baker^[1] to give a simultaneous approximation theorem to the values of exponential function at algebraic points, which slightly sharpens and generalises the results of Chudnovsky^[2],[3]. This work is supported by the National Natural Science Foundation of China.     

Rational points of algebraic curves

It is proved that the number of k-points of certain curves of genus g>1 is finite.     

Stable points on algebraic stacks

This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.     

Nonstandard points on algebraic curves

Let Γ be an algebraic curve which is given by an equation f(x, y) = 0, f(x, y) ∈ k[x, y] where k is an algebraic number field and f(x, y) is irreducible. Suppose that there exists an ${}^1\text{Γ}$ a nonstandard point $\text{(ξ, η) ∈}{}^1\text{k ×}{}^1\text{k}$. Then k(ξ, η) is (isomorphic to) the algebraic function field of Γ and, at the same time, is a subfield of ${}^1\text{k}$. Correlating the divisors of the function field k(ξ, η) and of the number field ${}^1\text{k}$, we develop an analogue of the Artin-Whaples theory of the product formula. This leads to one of Siegel's basic inequalities for rational points on algebraic curves.     

Separators of points on algebraic surfaces

For a finite set of points X⊆Pⁿ and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pⁿ, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P² in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.     

Solution of an algebraic equation using an irrational iteration function

It is proved that, for the choice z _n ^[n] = ?a ₁ of the initial approximation, the sequence of approximations z _n ^[i+1] = φ _n (z _n ^[i] ), [i] = 0, 1, 2, ..., of a solution of every canonical algebraic equation with real positive roots which is of the form $$P_n (z) = z^n + a_1 z^{n - 1} + a_2 z^{n - 2} + \ldots + a_n = 0, n = 1,2, \ldots ,$$ where the sequence is generated by the irrational iteration function φ _n (z) = (z ⁿ ? P _n (z))^1/n , converges to the largest root z _n . Examples of numerical realization of the method for the problem of determining the energy levels of electron systems of a molecule or a crystal are presented. The possibility of constructing similar irrational iteration functions in order to solve an algebraic equation of general form is considered.     

Real intersection points of piecewise algebraic curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this work, we present an algorithm for computing the real intersection points of piecewise algebraic curves. It is primarily based on the interval zeros of the univariate interval polynomial in Bernstein form. An illustrative example is provided to show that the proposed algorithm is flexible.     

Approximation of a two-valued function by an algebraic polynomial

We consider the minimax model of a nonlinear structure for approximating a two-valued function by an algebraic polynomial. We establish optimality conditions as a strong generalization of P. L. Chebyshev alternance optimality conditions in approximation of a function by a polynomial.     

Critical points of polynomials

Acta Mathematica Hungarica - This paper is devoted to the problem of where the critical points of a polynomial are relative to their zeros. Classical and new developments are surveyed along with...     

Diagonal fixed points in algebraic recursion theory

The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by the Curry combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces. Supported in part by the Ministry of Education and Science of Republic of Bulgaria, contract No 705     

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve. Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG.     

On algebraic connectivity as a function of an edge weight

We consider the Laplacian matrix of a weighted graph, and how the algebraic connectivity, α, behaves when considered as a function of a single edge weight. Under suitable differentiability conditions, we bound the first derivative of α from above, show that α is necessarily concave down, and produce a lower bound on the second derivative of α. When α is simple, we discuss the effect of increasing an edge weight on the corresponding Fiedler vector. We also compute the limiting value of α as the edge weight increases to infinity.     

On the algebraic points of a definable set

This paper studies diophantine properties of sets definable in an o-minimal structure over the real field. The main theorem of the author’s recent paper with A. J. Wilkie is refined, and used to deduce a strong result on the density of algebraic points of such sets.      

On branching index of algebraic extension spectrum points

The paper studies the branching A-index and the topological branching index of points of the spectrum of Arens-Hoffman algebraic extension of a semisimple commutative Banach algebras A.