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Erratum to “Cyclic resultants” [J. Symbolic Comput. 39 (6) (2005) 653–669]
Journal of Symbolic Computation 40 (2005) 1126–1127 www.elsevier.com/locate/jsc
Erratum
Erratum to “Cyclic resultants” [J. Symbolic Comput. 39 (6) (2005) 653–669] Christopher J. Hillar∗ Department of Mathematics, University of California, Berkeley, CA 94720, United States Received 28 April 2005; accepted 3 May 2005 Available online 13 June 2005
1. Errata In Theorem 1.1 of Hillar (2005), a characterization was given for when two univariate polynomials share the same sequence of nonzero cyclic resultants. As pointed out to me by Rúa (2005), this description is partially incomplete. The corrected statement should be given as follows. Theorem 1.1. Let f and g be polynomials in C[x]. Then, f and g generate the same sequence of nonzero cyclic resultants if and only if there exist u, v ∈ C[x] with u(0) = 0 and nonnegative integers l1 , l2 such that deg(u) ≡ l2 − l1 (mod 2), and f (x) = (−1)l2 −l1 x l1 v(x)u(x −1 )x deg(u) g(x) = x l2 v(x)u(x). This change does not affect any of the other results in Hillar (2005). The cause for the missing case (when l1 ≡ l2 (mod 2)) stems from a minor miscalculation of the divisor of a certain rational function. For completeness, we state the correction here. All of the following equation references are taken from Hillar (2005). Let f = x l h ∈ C[x] in which h(0) = 0 and h has degree d. Then, from (3.2), the cyclic resultants of f are given by DOI of original article: 10.1016/j.jsc.2005.01.001.
∗ Tel.: +1 510 843 8424; fax: +1 510 642 8204.
E-mail address: [email protected]. 0747-7171/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsc.2005.05.001
C.J. Hillar / Journal of Symbolic Computation 40 (2005) 1126–1127
1127
(−1)l rm (h). Examining Eq. (3.4) following Corollary 3.3, it follows that the divisor of G d for f is given by the divisor of the rational function (−1)l ∞ ∞ zm zm exp − rm ( f ) rm (h) . = exp − m m m=1 m=1 Let α1 , . . . , αd be the roots of h. By the discussion for polynomials without roots of zero found at the beginning of Section 4, it follows that the divisor of G d for f is d
αi−1 − [1] . (−1)l a0−1 i=1
With this correction in hand, it is straightforward to modify the proof of Theorem 1.1 and derive the missing case in the characterization. References Hillar, C., 2005. Cyclic resultants. J. Symbolic Comput. 39, 653–669. Rúa, I.F., 2005. Private communication.
Erratum
Erratum to “Cyclic resultants” [J. Symbolic Comput. 39 (6) (2005) 653–669] Christopher J. Hillar∗ Department of Mathematics, University of California, Berkeley, CA 94720, United States Received 28 April 2005; accepted 3 May 2005 Available online 13 June 2005
1. Errata In Theorem 1.1 of Hillar (2005), a characterization was given for when two univariate polynomials share the same sequence of nonzero cyclic resultants. As pointed out to me by Rúa (2005), this description is partially incomplete. The corrected statement should be given as follows. Theorem 1.1. Let f and g be polynomials in C[x]. Then, f and g generate the same sequence of nonzero cyclic resultants if and only if there exist u, v ∈ C[x] with u(0) = 0 and nonnegative integers l1 , l2 such that deg(u) ≡ l2 − l1 (mod 2), and f (x) = (−1)l2 −l1 x l1 v(x)u(x −1 )x deg(u) g(x) = x l2 v(x)u(x). This change does not affect any of the other results in Hillar (2005). The cause for the missing case (when l1 ≡ l2 (mod 2)) stems from a minor miscalculation of the divisor of a certain rational function. For completeness, we state the correction here. All of the following equation references are taken from Hillar (2005). Let f = x l h ∈ C[x] in which h(0) = 0 and h has degree d. Then, from (3.2), the cyclic resultants of f are given by DOI of original article: 10.1016/j.jsc.2005.01.001.
∗ Tel.: +1 510 843 8424; fax: +1 510 642 8204.
E-mail address: [email protected]. 0747-7171/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsc.2005.05.001
C.J. Hillar / Journal of Symbolic Computation 40 (2005) 1126–1127
1127
(−1)l rm (h). Examining Eq. (3.4) following Corollary 3.3, it follows that the divisor of G d for f is given by the divisor of the rational function (−1)l ∞ ∞ zm zm exp − rm ( f ) rm (h) . = exp − m m m=1 m=1 Let α1 , . . . , αd be the roots of h. By the discussion for polynomials without roots of zero found at the beginning of Section 4, it follows that the divisor of G d for f is d
αi−1 − [1] . (−1)l a0−1 i=1
With this correction in hand, it is straightforward to modify the proof of Theorem 1.1 and derive the missing case in the characterization. References Hillar, C., 2005. Cyclic resultants. J. Symbolic Comput. 39, 653–669. Rúa, I.F., 2005. Private communication.