Handbook of theoretical computer science (vol. A)
Complexity and real computation
Complexity and real computation
Some Remarks on the L-Conjecture
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
The Complexity of Factors of Multivariate Polynomials
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Algebraic Complexity Theory
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It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n!) by straight-line programs. Formally, we say that a sequence of integers an is ultimately f(n)-computable, if there exists a nonzero integer sequence mn such that for any n, anmn can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f(n). Shub and Smale [12] showed that if n! is ultimately hard to compute, then the algebraic version of NP ≠ P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in a short interval, a subexponential upper bound (exp(c√log n log log n)) for the ultimate complexity of n! is proved in this paper, and a randomized subexponential algorithm constructing such a short straight-line program is presented as well.