Polynomial recurrences and cyclic resultants

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 fullsequence of cyclic resultants; however, the known techniques proving thisresult give no effective computational bounds. We prove that a generic monicpolynomial of degree $d$ is determined by its first $2^{d+1}$ cyclic resultantsand that a generic monic reciprocal polynomial of even degree $d$ is determinedby its first $2\cdot 3^{d/2}$ of them. In addition, we show that cyclicresultants satisfy a polynomial recurrence of length $d+1$. This result givesevidence supporting the conjecture of Sturmfels and Zworski that $d+1$resultants determine $f$. In the process, we establish two general results ofindependent interest: we show that certain Toeplitz determinants are sufficientto determine whether a sequence is linearly recurrent, and we give conditionsunder which a linearly recurrent sequence satisfies a polynomial recurrence ofshorter length.