Polynomial recurrences and cyclic resultants

Details Polynomial recurrences and cyclic resultants Polynomial recurrences and cyclic resultants

2004-11-18

arxiv.org/abs/math/0411414v4

Mathematics

Algebraic Geometry

Proceedings of the AMS

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Polynomial recurrences and cyclic resultants does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Polynomial recurrences and cyclic resultants, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Polynomial recurrences and cyclic resultants will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-258382