Hilbert's Nullstellensatz is in the polynomial hierarchy
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
A weak version of the Blum, Shub, and Smale model
Journal of Computer and System Sciences - Special issue: dedicated to the memory of Paris Kanellakis
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Complexity and real computation
Complexity and real computation
Straight-line complexity and integer factorization
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Hilbert''s Nullstellensatz is in the Polynomial Hierarchy
Hilbert''s Nullstellensatz is in the Polynomial Hierarchy
Faster p-adic feasibility for certain multivariate sparse polynomials
Journal of Symbolic Computation
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In this paper, we show several connections between the L-conjecture, proposed by Bürgisser, and the boundedness theorem for the torsion points on elliptic curves. Assuming the WL-conjecture, which is a much weaker version of the L-conjecture, a sharper bound is obtained for the number of torsion points over extensions of k on an elliptic curve over a number field k, which improves Masser's result. It is also shown that the Torsion Theorem for elliptic curves follows directly from the WL-conjecture. Since the current proof of the Torsion Theorem for elliptic curves uses considerable machinery from arithmetic geometry, and the WL-conjecture differs from the trivial lower bound only at a constant factor, these results provide an interesting example where increasing the constant factor in a trivial lower bound of straight-line complexity is very difficult. Our results suggest that the Torsion Theorem may be viewed as a lower bound result in algebraic complexity, and a lot can be learned from the proof of the Uniform Boundedness Theorem to construct the proofs of the WL-conjecture or even the L-conjecture.