形式幂级数与分次代数的Veronese构造
The Veronese Construction for Formal Power Series and Graded Algebras
摘要：
设$(a_n)_{n \geq 0}$是一个复数序列，使得它的生成级数对某个多项式$h(t)$满足$\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$.对于任意的$r \geq 1$，我们研究了$h(t)$的系数级数到$h^{< r >}(t)$的系数级数的变换，其中$\sum_{n \geq 0} a_{nr} t^n = \frac{h^{< r >}(t)}{(1-t)^d}$ .我们给出了一个精确的描述，这种转变，并表明，在一些自然温和的假设下，$h^{< r >}(t)$的根收敛时，$r$走向无穷大。特别地，如果$\sum_{n \geq 0} a_n t^n$是标准分次$k$ -代数$A$的希尔伯特级数，则这成立。另外，如果$A$是Cohen-Macaulay，则$h^{< r >}(t)$的系数随$r$单调增加。如果$A$是单纯复$Δ$的Stanley-Reisner环，则这与$Δ$的$r$次边向细分有关，这又允许相应$f$ -向量的行为的一些推论。
英文摘要：
Let $(a_n)_{n \geq 0}$ be a sequence of complex numbers such that its generating series satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \geq 1$ we study the transformation of the coefficient series of $h(t)$ to that of $h^{< r >}(t)$ where $\sum_{n \geq 0} a_{nr} t^n = \frac{h^{< r >}(t)}{(1-t)^d}$. We give a precise description of this transformation and show that under some natural mild hypotheses the roots of $h^{< r >}(t)$ converge when $r$ goes to infinity. In particular, this holds if $\sum_{n \geq 0} a_n t^n$ is the Hilbert series of a standard graded $k$-algebra $A$. If in addition $A$ is Cohen-Macaulay then the coefficients of $h^{< r >}(t)$ are monotonely increasing with $r$. If $A$ is the Stanley-Reisner ring of a simplicial complex $Δ$ then this relates to the $r$th edgewise subdivision of $Δ$ which in turn allows some corollaries on the behavior of the respective $f$-vectors.
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