Regular sequences of symmetric polynomials
Regular sequences of symmetric polynomials
复制标题
DOI：
10.4171/rsmup/121-11
复制
发表时间：
2008-01
期刊：
影响因子：
--
通讯作者：
A. Conca;C. Krattenthaler;J. Watanabe
中科院分区：
文献类型：
--
作者：
A. Conca;C. Krattenthaler;J. Watanabe
用p_k表示n元对称多项式的k次幂和。将二项式系数的q-模拟解释为希尔伯特函数使我们发现n个变量的n个连续幂和形成一个正则序列。然后，我们考虑以下问题：描述子集n幂和形成一个正规序列。一个必要条件是n！除以元素度数的乘积。要找到一个容易验证的充分条件原来是令人惊讶的困难已经在3个变量。给定正整数a<B<c且GCD（a，B，c）=1，我们猜想p_a，p_B，p_c是正则序列当且仅当6整除abc.我们通过几个特殊的例子证明了这个猜想。
Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.