The Makar-Limanov Algebraically Closed Skew Field

We re-prove the Makar-Limanov theorem on the existence of an algebraically closed skew field in the sense of there being a solution for any (generalized) polynomial equation. A new example of such a skew field is presented in which the Makar-Limanov construction is contained as a skew subfield. Our reasoning is underpinned by the main ideas of the original proof, but we employ a simpler argument for proving that the skew field constructed is algebraically closed.     

Algebraically closed noncommutative polynomial rings

Let R be a noncommutative polynomial ring over the division ring K where K has center F. Then R = K[x,σ,D]where σ is a monomorphism of K and D is a σ-derivaton K. R is called dimension finite if (K: F_σ)<∞ and (K: F_D)<∞ where F_σ is the subfield of F fixed under σand F_D is the subfied of F of D-constants. R is algebraically closed if every nonconstant polynomial in Rfactors completely into linear factors. The algebraically closed dimension finite polynomial rings are determined. s done by reducing the problem to two classes: skew polynomial rings and differential polynomial rings. Examples algebraically closed polynomial rings which are not dimensfinite are given.     

环的代数封闭性

证明了任一环有代数封闭的扩张环,且实封闭域上的四元数体是代数封闭的,给出了代数封闭环的若干性质.     

Hopf Algebras on Schurian Quivers

Let H be a Hopf algebra with a finite-dimensional, nontrivial space of skew primitive elements, over an algebraically closed field of characteristic zero. We prove that H contains either the polynomial algebra as a Hopf subalgebra, or a certain Schurian simple-pointed Hopf subalgebra. As a consequence, a complete list of the locally finite, simple-pointed Hopf algebras is obtained. Also, the graded automorphism group of a Hopf algebra on a Schurian Hopf quiver is determined, and the relation between this group and the automorphism groups of the corresponding Hopf quiver, is clarified.     

Surjectivity of the resultant map: a solution to the inverse bezout problem

Let K be an algebraically closed field of characteristic zero or a real closed field. We prove that over the field K every polynomial in one variable of degree mn is the resultant of two polynomials in two variables of degrees m and n     

Polynomial Identities and Nonidentities of Split Jordan Pairs

We show that split Jordan pairs over rings without 2-torsion can be distinguished by polynomial identities with integer coefficients. In particular, this holds for simple finite-dimensional Jordan pairs over algebraically closed fields of characteristic not 2. We also generalize results of Drensky and Racine and of Rached and Racine on polynomial identities of, respectively, Jordan algebras and Jordan triple systems.     

The Linear Coordinate Preserving Problem

We prove that every K-endomorphism of a rank 2 polynomial algebra over an algebraically closed field K of positive characteristic taking all linear coordinates to coordinates is an automorphism. We give a new characterization of coordinates of K[t][x, y], where K is an algebraically closed field of any characteristic. We also explore the close connection between coordinates and permutation polynomials of finite fields.     

Complexity of “wild” matrix problems and of isomorphism of algebras and graphs

Journal of Mathematical Sciences - It is shown that isomorphism of semisimple algebras over an algebraically closed field is recognized in polynomial time. The polynomial equivalence of isomorphism...     

2-齐次多项式集合的吴消元法

1引言 考虑2-齐次多项式集合p,两组变量分别记为x=(x_1…,x_n)与y=(y_1…,y_m),p的零点集Zero(P)为有限点集.利用吴消无法([7],[6])可以将此类多项式集合完全三角化从而求得全部解,但这样消元的复杂度甚高.     

Extremal Valued Fields

It is shown that every finite-dimensional skew field whose center is an extremal valued field is defect free. We construct an example of an algebraically complete valued field such that a finite-dimensional skew field over it has a non-trivial defect, that is, there exist algebraically complete valued fields that are not extremal.     

Simply connected algebras of polynomial growth

The representation theory of polynomial growth strongly simply connected finite dimensional algebras over an algebraically closed field is established.     

Stably Calabi-Yau algebras and skew group algebras

Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.     

On the fundamental theorem of algebra for polynomial equations over real composition algebras

A topological proof is given that real compositions algebras of finite dimension greater than one are algebraically closed under polynomial equations with a tame tail.     

Algebraically closed structures in positive logic

In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.     

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

Counting solutions of polynomial systems via iterated fibrations

The paper introduces a new polynomial to count the solutions of a system of polynomial equations and inequations over an algebraically closed field of characteristic zero based on the triangular decomposition algorithm by J. M. Thomas of the nineteen-thirties. In the special case of projective varieties examples indicate that it is a finer invariant than the Hilbert polynomial. Received: 8 March 2008; Revised: 12 August 2008     

Identities of *-superalgebras and almost polynomial growth

We study the growth of the codimensions of a *-superalgebra over a field of characteristic zero. We classify the ideals of identities of finite dimensional algebras whose corresponding codimensions are of almost polynomial growth. It turns out that these are the ideals of identities of two algebras with distinct involutions and gradings. Along the way, we also classify the finite dimensional simple *-superalgebras over an algebraically closed field of characteristic zero.     

Radicals in skew polynomial and skew Laurent polynomial rings

We provide a general procedure for characterizing radical-like functions of skew polynomial and skew Laurent polynomial rings under grading hypotheses. In particular, we are able to completely characterize the Wedderburn and Levitzki radicals of skew polynomial and skew Laurent polynomial rings in terms of ideals in the coefficient ring. We also introduce the T-nilpotent radideals, and perform similar characterizations.     

Pure Skew Lattices in Rings

Given a ring $R$, let $S\subseteq R$ be a pure multiplicative band that is closed under the cubic join operation $x\nabla y = x+y+yx-xyx-yxy.$ We show that $\left( S,\cdot,\nabla\right) $ forms a pure skew lattice if and only if $S$ satisfies the polynomial identity $\left(xy-yx\right)^{2}z = z\left(xy-yx\right)^{2}$. We also examine properties of pure skew lattices in rings.     

On the study of the decomposition of matrix polynomials into factors

We present a survey of studies of questions on the decomposition of matrix polynomials into factors. The main attention is devoted to results obtained by P. S. Kazimirs 'kii on a method of solving the problem of determining the regular factors of a matrix polynomial over an algebraically closed field of characteristic zero. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.