论文标题
线性化多元偏度多项式的理论和应用
Theory and applications of linearized multivariate skew polynomials
论文作者
论文摘要
在这项工作中，引入了线性化的多元偏度多项式对除法环的引入。此类多项式在相应的centralizer上是正确的线性，并在有限场，组环或差分多项式环上概括了线性化的多项式环。它们的自然评估与基于其余的自由多元偏度多项式的评估有关。结果表明，当分区为结合类别时，p独立的集合是由正确的线性独立集给出的集合。因此，有限生成的p封闭组对应于有限维右矢量空间的列表，从而扩展了LAM和LEROY在单变量偏度多项式上的结果。还表明，自由偏度多项式的产物转化为线性化的多元偏度多项式的坐标组成，进而将其转化为相应的中央位置上的矩阵产物。后来，引入了线性化的多元Vandermonde矩阵，从而概括了多元Vandermonde，Moore和Wronskian矩阵。先前的结果显式地给出了他们的排名。然后引入了除法环的p-galois扩展，从而概括了经典（有限的）galois扩展。三个GALOIS理论结果被推广到此类扩展：Artin的延长度，Galois通信和Hilbert的定理90。
In this work, linearized multivariate skew polynomials over division rings are introduced. Such polynomials are right linear over the corresponding centralizer and generalize linearized polynomial rings over finite fields, group rings or differential polynomial rings. Their natural evaluation is connected to the remainder-based evaluation of free multivariate skew polynomials. It is shown that P-independent sets are those given by right linearly independent sets when partitioned into conjugacy classes. Hence finitely generated P-closed sets correspond to lists of finite-dimensional right vector spaces, extending Lam and Leroy's results on univariate skew polynomials. It is also shown that products of free multivariate skew polynomials translate into coordinate-wise compositions of linearized multivariate skew polynomials, which in turn translate into matrix products over the corresponding centralizers. Later, linearized multivariate Vandermonde matrices are introduced, which generalize multivariate Vandermonde, Moore and Wronskian matrices. The previous results explicitly give their ranks in general. P-Galois extensions of division rings are then introduced, which generalize classical (finite) Galois extensions. Three Galois-theoretic results are generalized to such extensions: Artin's theorem on extension degrees, the Galois correspondence and Hilbert's Theorem 90.
