Generic Polynomials with Few Parameters
Generic Polynomials with Few Parameters
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参数很少的泛型多项式
DOI：
10.1006/jsco.1999.0385
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Elena Mattig
中科院分区：
文献类型：
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作者：
G. Kemper;Elena Mattig
我们称多项式g（t1，. . .，tm，X）在域K上对群G是泛属的，如果它有Galois群G作为X中的多项式，且如果每个Galois域扩张N/L有K？L和Gal（N/L）？G作为一个合适的专门化g（？）1，. . . , ? m，X）与？我？L.我们讨论如何的合理性的忠实的线性表示的不变字段导致一个通用的多项式，这往往是特别简单，因此是有用的。然后，我们考虑各种例子和应用的特征0和积极的特征。这些结果包括所谓的向量多项式和推广的嵌入标准Abhyankar。给出了上幂幺矩阵群和上三角矩阵群的定义特征域上的一般多项式的递推公式，以及群Gu ~ 2（q ~ 2）和Go ~ 3（q）的一般多项式的显式公式.
We call a polynomial g(t1, . . . , tm,X ) over a field K generic for a group G if it has Galois group G as a polynomial inX , and if every Galois field extension N/L withK?L and Gal(N/L) ?G arises as the splitting field of a suitable specializationg (?1, . . . , ?m, X) with?i?L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU2(q2) andGO3 (q).