Faster p-adic Feasibility for Certain Multivariate Sparse Polynomials

TL;DR

This paper introduces new algorithms for detecting p-adic rational roots of sparse polynomials, achieving sub-exponential complexity in certain cases and establishing NP-hardness for some multivariate cases.

Contribution

It provides the first sub-exponential algorithms for specific classes of sparse polynomials and proves NP-hardness for detecting p-adic roots in multivariate cases.

Findings

Sub-exponential detection algorithms for certain multivariate polynomials

NP membership for honest (n+1)-nomials under specific conditions

NP-hardness of detecting p-adic roots for multivariate sparse polynomials

Abstract

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.