The $m$-th cyclic resultant of a polynomial $f \in \mathbb C[x]$ is \[r_m = \text{Res}(f,x^m-1).\] We characterize polynomials having the same set of nonzero cyclic resultants. Sequences of the form $r_m$ arise (among many other places) as the cardinalities of sets of periodic points for toral endomorphisms. Let $A$ be a $d$-by-$d$ integer matrix and let $X = \mathbb T^{d} = \mathbb R^{d}/\mathbb Z^{d}$ denote the $d$-dimensional additive torus. Then, $A$ acts on $X$ by multiplication mod $1$; that is, it defines a map $T: X \to X$ given by \[T(\mathbf{x}) = A\mathbf{x} \mod \mathbb Z^d.\]
Let $\text{Per}_m(T) = \{\mathbf{x} \in \mathbb T^{d} : T^m(\mathbf{x}) = \mathbf{x}\}$ be the set of points fixed under the map $T^m$. Under the ergodicity condition that no eigenvalue of $A$ is a root of unity, it turns out that \[|\text{Per}_m(T)| = |\det(A^m-\text{I})| = |r_m(f)|,\] in which $f$ is the characteristic polynomial of $A$.
We describe how our results allow for reconstruction of such dynamical systems from their zeta functions, \[Z(T,z) = \exp{\left(- \sum_{m = 1}^\infty { |\text{Per}_m(T)|\frac{z^m}{m} } \right)}.\] Part of this work is joint with L. Levine (U.C. Berkeley).
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