Sturmfels–Zworski conjecture on reconstruction from cyclic resultants

Let $f(x)\in\mathbb C[x]$ be a generic monic polynomial of degree $d$. Its cyclic resultants are the quantities associated to $f$ obtained by evaluating the resultant with $x^m-1$ for positive integers $m$. A polynomial is reciprocal when its coefficients satisfy the corresponding palindromic symmetry; here the reciprocal polynomial is allowed to be non-monic and has even degree $d$.

Sturmfels–Zworski conjecture. A generic monic polynomial $f(x)\in\mathbb C[x]$ of degree $d$ is determined by its first $d+1$ cyclic resultants. Moreover, if $f$ is non-monic reciprocal of even degree $d$, then the number of resultants needed for inversion is $d/2+2$.

This conjecture concerns finite reconstruction of polynomials from cyclic resultants, which encode periodic-point data in dynamical systems and are closely related to zeta functions. The paper notes that $2^{d+1}$ resultants are known to suffice, while empirical evidence suggests that the conjectured bounds are substantially tighter.