@article{1856d6e7-70bc-340c-b969-e22e9bef8429,
    issn = {00029939, 10886826},
    url = {http://www.jstor.org/stable/20534741},
    abstract = {Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th 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.},
    author = {Christopher J. Hillar and Lionel Levine},
    journal = {Proceedings of the American Mathematical Society},
    number = {6},
    pages = {1607--1618},
    publisher = {American Mathematical Society},
    title = {Polynomial Recurrences and Cyclic Resultants},
    urldate = {2025-04-16},
    volume = {135},
    year = {2007},
    date-added = {2025-4-16 13:55:50 +0100}
}

@article{1856d6e7-70bc-340c-b969-e22e9bef8429, issn = {00029939, 10886826}, url = {http://www.jstor.org/stable/20534741}, abstract = {Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th 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.}, author = {Christopher J. Hillar and Lionel Levine}, journal = {Proceedings of the American Mathematical Society}, number = {6}, pages = {1607--1618}, publisher = {American Mathematical Society}, title = {Polynomial Recurrences and Cyclic Resultants}, urldate = {2025-04-16}, volume = {135}, year = {2007}, date-added = {2025-4-16 13:55:50 +0100} }