关于因子分解不变量与Hilbert函数
On factorization invariants and Hilbert functions
摘要：
可消交换半群中的非唯一因子分解经常使用组合因子分解不变量来研究，组合因子分解不变量赋予每个半群元素一个由因子分解结构确定的量。对于数值半群（自然数的加法子半群），已知几个因子分解不变量允许足够大的半群元素的可预测行为。特别地，悬链线度和δ集合不变量都是最终周期性的，并且Ω素性不变量是最终拟线性的。在本文中，我们证明了这些不变量是如何确定的分次模的Hilbert函数。在这样做时，我们将上述每个最终的行为结果扩展到非线性生成半群，并提供了一个新的框架，通过它来研究在这种情况下的因子分解结构。
英文摘要：
Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical semigroups (additive subsemigroups of the natural numbers), several factorization invariants are known to admit predictable behavior for sufficiently large semigroup elements. In particular, the catenary degree and delta set invariants are both eventually periodic, and the omega-primality invariant is eventually quasilinear. In this paper, we demonstrate how each of these invariants is determined by Hilbert functions of graded modules. In doing so, we extend each of the aforementioned eventual behavior results to finitely generated semigroups, and provide a new framework through which to study factorization structures in this setting.
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