Factor Algebras of Free Algebras: On a Problem of G. Bergman

Let A_n = K x₁,...,x_n be a free associative algebra over a fieldK. In this paper, examples are given of elements u A_n, n 3,such that the factor algebra of A_n over the ideal generatedby u is isomorphic to A_n–1, and yet u is not a primitiveelement of A_n (that is, it cannot be taken to x₁ by an automorphismof A_n). If the characteristic of the ground field K is 0, thisyields a negative answer to a question of G. Bergman. On theother hand, by a result of Drensky and Yu, there is no suchexample for n = 2. It should be noted that a similar questionfor polynomial algebras, known as the embedding conjecture ofAbhyankar and Sathaye, is a major open problem in affine algebraicgeometry. 2000 Mathematics Subject Classification 16S10, 16W20(primary); 14A05, 13B25 (secondary).     

Tame and Wild Coordinates of

Let be a field of characteristic zero. We characterize coordinates and tame coordinates in , i.e. the images of respectively under all automorphisms and under the tame automorphisms of . We also construct a new large class of wild automorphisms of which maps to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of .

     

Lifting of polynomial symplectomorphisms and deformation quantization

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the a?ne space and is closely related to several major open problems in algebraic geometry and ring theory.

This paper is a continuation of the study [19 Kanel Belov, A., Razavinia, F., Zhang, W. (2017). Bergman’s centralizer theorem and quantization. Commun. Algebra 1–7.[Taylor &; Francis Online] , [Google Scholar]].     

基于四波混频过程和线性分束器产生四组份纠缠

多组份纠缠是量子信息处理的重要资源,它的产生通常涉及到许多复杂的线性和非线性过程.本文从理论上提出了一种利用两个独立的四波混频过程和线性分束器产生真正的四组份纠缠的方案,其中,线性分束器的作用是将两个独立的四波混频过程联系起来.首先应用部分转置正定判据研究了强度增益对四组份纠缠的影响,结果表明,在整个增益区域内都存在真正的四组份纠缠,并且随着强度增益的增加,纠缠也在增强.然后研究了线性分束器的透射率对四组份纠缠的影响,发现只要线性分束器的透射率不为0或1,该系统也可以产生真正的四组份纠缠.最后,通过研究该系统可能存在的三组份纠缠和两组份纠缠来揭示该系统的纠缠结构.本文理论结果为实验上利用原子系综四波混频过程产生真正的四组份纠缠提供了可靠的方案.     

On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) $K_1$ and $K_2$ over a field k, a k-isomorphism of $K_1[t]$ ($K_1(t)$) and $K_2[t]$ ($K_2(t)$) implies a k-isomorphism of $K_1$ and $K_2$. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose $K(t)?k(t_1,t_2,t_3)$, then $K?k(t_1,t_2)$.2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let $A=K[x,y,z,w]/M$ be the coordinate ring of M. Suppose $A[t]?k[x_1,x_2,x_3,x_4]$, then $\mathrm{frac}(A)?k(x_1,x_2,x_3)$, where $\mathrm{frac}(A)$ is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171].     

Quantum State Sharing by Using Two-Mode and Single-Mode Squeezed State Lights

We propose a quantum state sharing scheme for continuous variables using bright two-mode squeezed state and single-mode squeezed state light. The squeezing of a single-mode state is applied to enhance the security of information in quantum teleportation network. The signal-to-noise ratio of communication and the fidelity between the secret and reconstruction state are analysed. It is shown that both the receivers of Bob and Charlie cannot extract information with a high signal-to-noise ratio because of the large noise come from the other quadrature component of single mode squeezed state. Anyone of Bob and Charlie can retrieve the quantum state with a high signal-to-noise ratio if and only if the other one cooperates with the measurement.     

Automorphic orbit problem for polynomial algebras

It is proved that every endomorphism preserving the automorphic orbit of a non-trivial element of the rank two polynomial algebra over the complex number field is an automorphism.     

On the density of the set of generators of a polynomial algebra

Let be the polynomial algebra over a field of characteristic . We call a polynomial coordinate (or a generator) if for some polynomials . In this note, we give a simple proof of the following interesting fact: for any polynomial of the form where is a polynomial without constant and linear terms, and for any integer , there is a coordinate polynomial such that the polynomial has no monomials of degree . A similar result is valid for coordinate -tuples of polynomials, for any . This contrasts sharply with the situation in other algebraic systems.

On the other hand, we establish (in the two-variable case) a result related to a different kind of density. Namely, we show that given a non-coordinate two-variable polynomial, any sufficiently small perturbation of its non-zero coefficients gives another non-coordinate polynomial.

     

Polynomial Retracts and the Jacobian Conjecture

Let be the polynomial algebra in two variables over a field of characteristic . A subalgebra of is called a retract if there is an idempotent homomorphism (a retraction, or projection) such that The presence of other, equivalent, definitions of retracts provides several different methods of studying and applying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.