Polynomial Retracts and the Jacobian Conjecture
Polynomial Retracts and the Jacobian Conjecture
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DOI：
10.1090/s0002-9947-99-02251-5
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发表时间：
1997-01
影响因子：
1.3
通讯作者：
V. Shpilrain;Jietai Yu
中科院分区：
文献类型：
--
作者：
V. Shpilrain;Jietai Yu
施皮林与余杰泰设K[x，y]是特征为0的域K上的二元多项式代数。K[x，y]的子代数R称为收缩子代数，如果存在幂等同态（收缩或投影）<$：K[x，y] → K[x，y]使得<$（K[x，y]）= R.其他等价的缩进定义的存在提供了几种不同的研究方法，并汇集了来自组合代数、同调代数和代数几何的思想。本文刻画了K[x，y]到一个自同构的所有收缩，并给出了这一刻画的几个应用，特别是对著名的Jacobian猜想的应用。值得注意的是，我们证明了如果K[x，y]的多项式映射具有可逆的雅可比矩阵并且固定一个非常数多项式，则是自同构。
VLADIMIR SHPILRAIN AND JIE-TAI YUAbstract. Let K[x,y] be the polynomial algebra in two variables over a fieldK of characteristic 0. A subalgebra R of K[x,y] is called a retract if there is anidempotent homomorphism (a retraction, or projection) ϕ : K[x,y] → K[x,y] suchthat ϕ(K[x,y]) = R. The presence of other, equivalent, definitions of retractsprovides several different methods of studying them, and brings together ideasfrom combinatorial algebra, homological algebra, and algebraic geometry. In thispaper, we characterize all the retracts of K[x,y] up to an automorphism, andgive several applications of this characterization, in particular, to the well-knownJacobian conjecture. Notably, we prove that if a polynomial mapping ϕ of K[x,y]has invertible Jacobian matrix and fixes a non-constant polynomial, then ϕ is anautomorphism.