ABSTRACT
Abstract We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In particular, we show that while () is doable in quantum randomized polynomial time when m=2, () is nearly NP-complete for general m. In particular, () is in NP for most inputs and, under a plausible hypothesis involving primes in arithmetic progression (implied by the generalized Riemann hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for () would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to be new. As a consequence we can also address recent questions on the complexity of polynomial factorization posed by Cox, and by Karpinski and Shparlinski.
