Cyclotomic Identity Testing and Applications

We consider the cyclotomic identity testing (CIT) problem: given a polynomial f(x1,…,xk), decide whether f(ζne1, …,ζnek) is zero, where ζn = e2π i/n is a primitive complex n-th root of unity and e1,…,ek are integers, represented in binary. When f is given by an algebraic circuit, we give a randomized polynomial-time algorithm for CIT assuming the generalised Riemann hypothesis (GRH), and show that the problem is in NP unconditionally. When f is given by a circuit of polynomially bounded degree, we give a randomized NC algorithm. In case f is a linear form we show that the problem lies in NC. Towards understanding when CIT can be solved in deterministic polynomial-time, we consider so-called diagonal depth-3 circuits, i.e., polynomials f ∑mi=1 g+idi, where gi is a linear form and di a positive integer given in unary. We observe that a polynomial-time algorithm for CIT on this class would yield a sub-exponential-time algorithm for polynomial identity testing. However, assuming GRH, we show that if the linear forms gi are all identical then CIT can be solved in polynomial time. Finally, we use our results to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in randomized NC.

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