V. V. Bavula
Explicit generators are found for the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in $n$ variables. An analogue of the polynomial Jacobian homomorphism is found.
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Explicit generators are found for the group $G_2$ of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that $$ G_2\simeq S_2\ltimes \mT^2\ltimes \Z\ltimes ((K^*\ltimes E_\infty (\mS_1))\boxtimes_{\GL_\infty (K)}(K^*\ltimes E_\infty (\mS_1)))$$ where $S_2$ is the symmetric group, $\mT^2$ is the 2-dimensional torus, $E_\infty (\mS_1)$ is the subgroup of $\GL_\infty (\mS_1)$ generated by the elementary matrices. In the proof, we use and prove several results on the index of operators, and…
The algebra $\mS_n$ in the title is obtained from a polynomial algebra $P_n$ in $n$ variables by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of $P_n$. Ignoring non-Noetherian property, the algebra $\mS_n$ belongs to a family of algebras like the Weyl algebra $A_n$ and the polynomial algebra $P_{2n}$. The group of automorphisms $G_n$ of the algebra $\mS_n$ is found: $$ G_n=S_n\ltimes \mT^n\ltimes \Inn (\mS_n) \supseteq S_n\ltimes \mT^n\ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}=:G_n' $$ where $S_n$ is the sym…
We study in detail the %Shrek algebra $\mS_n$ in the title which is an algebra obtained from a polynomial algebra $P_n$ in $n$ variables by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of $P_n$. The algebra $\mS_n$ is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals {\em commute}. It is proved that the classical Krull dimension of $\mS_n$ is $2n$; but the weak and the global dimensions of $\mS_n$ are $n$. The prime and maximal spectra of $\mS_n$ are found, and the simple $\mS_n$-modules…
An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$
Inverse limits of automorphisms of truncated polynomials and applications related to Jacobian conjecture
本文利用特征 $p$ 多项式环与逆极限技术，证明满足 Jacobian 条件且保持极大理想的多项式环同态必为自同构，为 Jacobian 猜想提供新结论。
The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.
The algebra $\mS_n$ of one-sided inverses of a polynomial algebra $P_n$ in $n$ variables is obtained from $P_n$ by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of the algebra $P_n$. The algebra $\mS_n$ is a noncommutative, non-Noetherian algebra of classical Krull dimension $2n$ and of global dimension $n$ which is not a domain. If the ground field $K$ has characteristic zero then the algebra $\mS_n$ is canonically isomorphic to the algebra $K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n}, \int_1, ..., \int_n>$ of scalar integro-differential operato…
Automorphism groups of curves over arbitrary fields
本文证明任意有限群都可实现为任意域上某条曲线的自同构群，肯定解决函数域弱逆 Galois 问题，并将一般逆 Galois 问题归约到 K(T)。
In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. Also the class of Pascal finite polynomial automorphisms was introduced. Pascal finite polynomial maps constitute a generalization of exponential automorphisms to positive characteristic. In this note we explore properties of the algorithm while using Segre homotopy and reductions modulo prime number. We give a method of retrieving an inverse of a given polynomial automorphism $F$ with integer coefficients form a finite set of the inverses of its reductions modulo prime numbers. Some examples illu…
An application of a nonuniform global stability problem to the study of parametrized polynomial automorphisms
本研究提出参数化映射族的若干单射性定义，关联非自治微分系统全局非一致稳定性猜想，借此证明一类参数化多项式自同构在特定参数下有多项式逆，与雅可比猜想相关。
Given a polynomial endomorphism F of the n-dimensional affine space over a field K, we define a sequence of polynomial endomorphisms of the affine space associated to F. We call F nice if there exists an integer m such that the m-th term of the sequence vanishes. Then F is invertible and its inverse may be computed in terms of the endomorphisms in the sequence. In this paper we study the class of nice polynomial automorphisms and obtain that this class is large and includes triangulable automorphisms and all linear cubic homogeneous polynomial automorphisms of nilpotence index up to 3. We prov…
Infinite transitivity of tame groups of automorphisms of affine spaces
该研究针对有限域上n维仿射空间的驯服自同构群，探究其对m个点有序子集的作用，构造可映射有序序列的驯服自同构并证明其复杂度的上下界，运用多数学分支方法获得相关定性与定量结果。
Let $k$ be a field of characteristic zero. Let $F = X + H$ be a polynomial mapping from $k^n \to k^n$, where $X$ is the identity mapping and $H$ has only degree two terms and higher. We say that the Jacobian matrix $JH$ of $H$ is strongly nilpotent with index $p$ if for all $X^{(1)},\ldots,X^{(p)} \in k^n$ we have \begin{align*} JH(X^{(1)})\ldots JH (X^{(p)}) = 0. \end{align*} Every $F$ of this form is a polynomial automorphism, i.e. there is a second polynomial mapping $F^{-1}$ such that $F \circ F^{-1} = F^{-1} \circ F = X$. We prove that the degree of the inverse $F^{-1}$ satisfies \begin{a…
The inverse Galois problem for connected algebraic groups
本文证明了任意基域上每个有限型连通群概形均可实现为某个射影几何整概形自同构群的单位分量，主要借助嵌入、等变紧化、爆破及Fitting理想等方法。
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