Cyclotomic Identity Testing and Applications

We consider the cyclotomic identity testing problem: given a polynomial $f(x_1,\ldots,x_k)$, decide whether $f(\zeta_n^{e_1},\ldots,\zeta_n^{e_k})$ is zero, for $\zeta_n = e^{2\pi i/n}$ a primitive complex $n$-th root of unity and integers $e_1,\ldots,e_k$. We assume that $n$ and $e_1,\ldots,e_k$ are represented in binary and consider several versions of the problem, according to the representation of $f$. For the case that $f$ is given by an algebraic circuit we give a randomized polynomial-time algorithm with two-sided errors, showing that the problem lies in BPP. In case $f$ is given by a circuit of polynomially bounded syntactic degree, we give a randomized algorithm with two-sided errors that runs in poly-logarithmic parallel time, showing that the problem lies in BPNC. In case $f$ is given by a depth-2 $\Sigma\Pi$ circuit (or, equivalently, as a list of monomials), we show that the cyclotomic identity testing problem lies in NC. Under the generalised Riemann hypothesis, we are able to extend this approach to obtain a polynomial-time algorithm also for a very simple subclass of depth-3 $\Sigma\Pi\Sigma$ circuits. We complement this last result by showing that for a more general class of depth-3 $\Sigma\Pi\Sigma$ circuits, a polynomial-time algorithm for the cyclotomic identity testing problem would yield a sub-exponential-time algorithm for polynomial identity testing. Finally, we use cyclotomic identity testing to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in coRNC: randomized NC with one-sided errors