Faster [formula omitted]-adic feasibility for certain multivariate sparse polynomials

Abstract

We present algorithms revealing new families of polynomials admitting sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we prove NP-completeness for the case of honest n-variate (n+1)-nomials and, for certain special cases with p exceeding the Newton polytope volume, constant-time complexity. 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 upper bounds for all 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.