Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
A course in computational algebraic number theory
A course in computational algebraic number theory
Algorithmic number theory
Counting curves and their projections
Computational Complexity
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Efficient p-adic cell decompositions for univariate polynomials
Journal of Complexity
Factoring Polynominals over p-Adic Fields
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Additive Complexity and Roots of Polynomials over Number Fields and \mathfrak{p} -adic Fields
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Counting Solutions to Equations in Many Variables over Finite Fields
Foundations of Computational Mathematics
Faster real feasibility via circuit discriminants
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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Relative to the sparse encoding, we show that deciding whether a univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a sharper complexity upper bound of P for polynomials with suitably generic p-adic Newton polygon. The best previous complexity upper bound was EXPTIME. We then prove an unconditional complexity lower bound of NP-hardness with respect to randomized reductions, for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss analogous results over R.