多项式递归式与循环结式
Polynomial recurrences and cyclic resultants
摘要：
设$K$是特征为零的代数闭域，令$f \in K[x]$ . $f$的第$m$个{ \it循环结式}是\[r_m = \text{Res}(f,x^m-1).\]。通用一元多项式由其循环结式的全序列确定;然而，证明该结果的已知技术没有给出有效的计算界限。我们证明了一个通用的一元多项式的次数$d$是由它的第一个$2^{d+1}$循环结式和通用的一元倒数多项式的次数$d$是由它的第一个$2\cdot 3^{d/2}$。此外，我们证明了循环结式满足长度为$d+1$的多项式递归。这一结果证明了Sturmfels和Zworski的猜想，即$d+1$的结式决定了$f$。在这个过程中，我们建立了两个独立的兴趣一般结果：我们表明，某些Toeplitz行列式是足以确定是否是线性递归序列，我们给出的条件下，线性递归序列满足多项式递归的长度较短。
英文摘要：
Let $K$ be an algebraically closed field of characteristic zero and let $f \in K[x]$. The $m$-th {\it cyclic resultant} of $f$ is \[r_m = \text{Res}(f,x^m-1).\] A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree $d$ is determined by its first $2^{d+1}$ cyclic resultants and that a generic monic reciprocal polynomial of even degree $d$ is determined by its first $2\cdot 3^{d/2}$ of them. In addition, we show that cyclic resultants satisfy a polynomial recurrence of length $d+1$. This result gives evidence supporting the conjecture of Sturmfels and Zworski that $d+1$ resultants determine $f$. In the process, we establish two general results of independent interest: we show that certain Toeplitz determinants are sufficient to determine whether a sequence is linearly recurrent, and we give conditions under which a linearly recurrent sequence satisfies a polynomial recurrence of shorter length.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
