Cyclic resultants

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Abstract

We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial f of degree d, there are exactly 2d1 distinct degree d polynomials with the same set of cyclic resultants as f. However, in the generic monic case, degree d polynomials are uniquely determined by their cyclic resultants. Moreover, two reciprocal (“palindromic”) polynomials giving rise to the same set of nonzero cyclic resultants are equal. In the process, we also prove a unique factorization result in semigroup algebras involving products of binomials. Finally, we discuss how our results yield algorithms for explicit reconstruction of polynomials from their cyclic resultants.

MSC

primary
11B83
14Q99
secondary
15A15
20M25

Keywords

Cyclic resultant
Binomial factorization
Group rings
Toral endomorphisms

Cited by (0)

This work is supported under a National Science Foundation Graduate Research Fellowship.