Linearized polynomial maps over finite fields
Linearized polynomial maps over finite fields
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有限域上的线性化多项式映射
DOI：
10.1016/j.jalgebra.2013.10.013
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Joost Berson
中科院分区：
文献类型：
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作者：
Joost Berson
我们考虑所谓的（多元）线性多项式描述的多项式映射。这些多项式是用一个固定的素数幂来定义的，比如q。线性化多项式没有混合项。考虑特征零域上不含混合项的可逆多项式映射，我们将只得到（直到变量的线性变换）三角映射，这是多项式自同构的最基本例子。然而，在有限域F上，由线性化多项式定义的q自同构（一般来说）具有完全不同的结构。也就是说，我们将证明F q上的线性多项式映射与F q上的一元多项式环中的系数矩阵一一对应。此外，多项式映射的合成转化为矩阵乘法，这意味着可逆线性多项式映射对应于可逆矩阵。线性化多项式自同构子群的这种交替描述导致了许多著名的代数（最著名的是雅可比猜想）的解决方案，这类多项式和多项式映射。
We consider polynomial maps described by so-called (multivariate) linearized polynomials. These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F q automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show that the linearized polynomial maps over F q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible linearized polynomial maps correspond to invertible matrices. This alternate description of the linearized polynomial automorphism subgroup leads to the solution of many famous conjectures (most notably, the Jacobian Conjecture) for this kind of polynomials and polynomial maps.