Faster p-adic Feasibility for Certain Multivariate Sparse Polynomials

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.

💡 Research Summary

The paper tackles the decision problem of whether a given sparse polynomial has a rational root in the p‑adic numbers (the FEAS‑p‑adic problem). Historically, this problem has been placed in EXPTIME or even higher complexity classes because the natural encoding size grows with the degree of the polynomial, which can be exponential in the number of terms. The authors adopt the sparse encoding model, where the input size is measured by the number of variables n, the number of non‑zero terms k, and the binary length of the coefficients and exponents. Within this framework they obtain several striking results that dramatically improve the known complexity bounds for important families of sparse polynomials.

First, they consider honest n‑variate (n + 1)‑nomials, i.e., polynomials in n variables that contain exactly n + 1 monomials with linearly independent exponent vectors. By analysing the Newton polytope of such a polynomial they show that the p‑adic valuation of the discriminant is controlled by the volume V of the polytope. If the prime p exceeds V and does not divide any coefficient, the polynomial is p‑adically non‑singular, and a single Hensel‑lifting step suffices to decide root existence. Consequently the decision can be made in constant time, independent of the input size. Moreover, they prove that for arbitrary primes p the problem for these polynomials lies in NP: a short certificate consisting of a candidate p‑adic integer together with a bound on its p‑adic valuation can be verified in polynomial time.

Second, the authors address univariate trinomials (three‑term polynomials). They invoke the theory of linear forms in p‑adic logarithms, which provides explicit lower bounds for expressions of the form |b₁ logₚα₁ + b₂ logₚα₂|ₚ. Using these bounds they show that any p‑adic root must lie in a bounded interval of the p‑adic integers whose size is polynomial in the input length. By enumerating this interval and checking each candidate with a fast Newton‑type iteration, they obtain an NP algorithm for the univariate trinomial case. This improves on the previous EXPTIME algorithms that relied on exhaustive search over exponentially many p‑adic digits.

Third, the paper establishes NP‑hardness of the univariate sparse FEAS‑p‑adic problem under randomized reductions. The reduction starts from a Boolean SAT instance and constructs a univariate sparse polynomial whose p‑adic root exists if and only if the SAT formula is satisfiable. A crucial ingredient is an efficient algorithm for producing a prime p that lies in a prescribed arithmetic progression (the progression’s modulus and offset are polynomially bounded in the size of the SAT instance). By adapting recent advances on the distribution of primes in arithmetic progressions (essentially a constructive version of Linnik’s theorem), the authors guarantee the existence of such a prime and find it in polynomial time with high probability. The resulting polynomial has O(m) terms, where m is the number of clauses, and its coefficients are small integers. The reduction succeeds with probability at least 2/3, which can be amplified by standard repetition. Hence detecting a p‑adic rational root for a univariate sparse polynomial is NP‑hard under randomized reductions.

Finally, the authors note that the smallest number of variables n for which the FEAS‑p‑adic problem becomes NP‑hard remains unknown. Their results settle the one‑variable case, but for n ≥ 2 only partial hardness results are known (e.g., for (n + 2)‑nomials). The paper suggests that extending the prime‑construction technique or developing new p‑adic discriminant bounds could close this gap.

In summary, the work introduces three major algorithmic breakthroughs: (1) constant‑time solvability for n‑variate (n + 1)‑nomials when p exceeds the Newton polytope volume and avoids the coefficients; (2) an NP algorithm for univariate trinomials based on linear forms in p‑adic logarithms; and (3) a randomized NP‑hardness proof for general univariate sparse polynomials via an efficient prime‑in‑progression construction. These contributions collectively shift the landscape of p‑adic feasibility from exponential to sub‑exponential (and even constant) time for large families of sparse polynomials, while also delineating the boundary where the problem becomes computationally intractable.