用素特征的Newton多项式生成对称场
Generation of the Symmetric Field by Newton Polynomials in prime Characteristic
摘要：
令$N_m = x^m + y^m$为$m \geq 1$的二元$m$次牛顿多项式。Dvornicich和Zannier证明了在特征零点上，三个牛顿多项式$N_a, N_b, N_c$总是足以生成$x$和$y$中的对称场，只要$a,b,c$是不同的正整数，使得$(a,b,c)=1$ .本文证明了在素数特征$p$的情况下，如果我们另外假设$a,b,c,a-b,a-c,b-c$与$p$是素数，则结果仍然成立。我们还提供了一个反例的情况下，其中一个假设是失踪。结果如下的研究广义范德蒙行列式在三个变量的因式分解，在一般假设下的因素作为产品的平凡范德蒙因子和一个不可约的因素。另一方面，反例与舒尔多项式因子作为线性因子的乘积的某些情况有关。
英文摘要：
Let $N_m = x^m + y^m$ be the $m$-th Newton polynomial in two variables, for $m \geq 1$. Dvornicich and Zannier proved that in characteristic zero three Newton polynomials $N_a, N_b, N_c$ are always sufficient to generate the symmetric field in $x$ and $y$, provided that $a,b,c$ are distinct positive integers such that $(a,b,c)=1$. In the present paper we prove that in case of prime characteristic $p$ the result still holds, if we assume additionally that $a,b,c,a-b,a-c,b-c$ are prime with $p$. We also provide a counterexample in the case where one of the hypotheses is missing. The result follows from the study of the factorization of a generalized Vandermonde determinant in three variables, that under general hypotheses factors as the product of a trivial Vandermonde factor and an irreducible factor. On the other side, the counterexample is connected to certain cases where the Schur polynomials factor as a product of linear factors.
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