Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Counting curves and their projections
Computational Complexity
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Complexity estimates depending on condition and round-off error
Journal of the ACM (JACM)
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
Polynomial factorization: a success story
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Counting Solutions to Equations in Many Variables over Finite Fields
Foundations of Computational Mathematics
Straight-line programs and torsion points on elliptic curves
Computational Complexity
Faster real feasibility via circuit discriminants
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Sub-linear root detection, and new hardness results, for sparse polynomials over finite fields
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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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.