几乎对称数值半群的Hilbert级数的对偶关系
Duality Relation for the Hilbert Series of Almost Symmetric Numerical Semigroups
摘要：
将几乎对称数值半群S（d ^m）与它的对偶H（d^m;z^{-1}）结合起来，得到了几乎对称数值半群S（d ^m;z）的Hilbert级数H（d^m;z）的对偶关系.在此基础上，建立了合合项的度的多重集与间隙F_j、生成元d_i及其线性组合的多重集之间的双射。分别给出了偶数和奇数指数的贝蒂数和的关系式。本文将对偶关系应用到具有最大嵌入维数的几乎对称半群的简单情形，给出了极小集d^m生成这类半群的充要条件.
英文摘要：
We derive the duality relation for the Hilbert series H(d^m;z) of almost symmetric numerical semigroup S(d^m) combining it with its dual H(d^m;z^{-1}). On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F_j, generators d_i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and efficient conditions for minimal set d^m to generate such semigroups.
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