  We present algorithms revealing new families of polynomials allowing
sub-exponential detection of p-adic rational roots, relative to the sparse
encoding. For instance, we show that the case of honest n-variate (n+1)-nomials
is doable in NP and, for p exceeding the Newton polytope volume and not
dividing any coefficient, in constant time. Furthermore, using the theory of
linear forms in p-adic logarithms, we prove that the case of trinomials in one
variable can be done in NP. The best previous complexity bounds for these
problems were EXPTIME or worse. Finally, we prove that detecting p-adic
rational roots for sparse polynomials in one variable is NP-hard with respect
to randomized reductions. The last proof makes use of an efficient construction
of primes in certain arithmetic progressions. The smallest n where detecting
p-adic rational roots for n-variate sparse polynomials is NP-hard appears to
have been unknown.
