On the intrinsic complexity of elimination theory
Journal of Complexity - Special issue: invited articles dedicated to J. F. Traub on the occasion of his 60th birthday
Computing over the reals with addition and order
Selected papers of the workshop on Continuous algorithms and complexity
Complexity and real computation
Complexity and real computation
Are lower bounds easier over the reals?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Lower Bounds Are Not Easier over the Reals: Inside PH
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
Straight-line complexity and integer factorization
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Valiant's model and the cost of computing integers
Computational Complexity
Two theorems on random polynomial time
SFCS '78 Proceedings of the 19th Annual Symposium on Foundations of Computer Science
VPSPACE and a transfer theorem over the complex field
Theoretical Computer Science
VPSPACE and a transfer theorem over the reals
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Decision versus evaluation in algebraic complexity
MCU'07 Proceedings of the 5th international conference on Machines, computations, and universality
VPSPACE and a transfer theorem over the complex field
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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We study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class VΠP0 as the set of families of polynomials that are exponential products of easily computable polynomials. We investigate the consequences of the hypothesis that these big products are themselves easily computable. For instance, this hypothesis would imply that the nonuniform versions of P and NP coincide. Our main result relates this hypothesis to Blum, Shub and Smale's algebraic version of P versus NP. Let K be a field of characteristic 0. Roughly speaking, we show that in order to separate PK from NPK using a problem from a fairly large class of “simple” problems, one should first be able to show that exponential products are not easily computable. The class of “simple” problems under consideration is the class of NP problems in the structure (K,+,–,=), in which multiplication is not allowed.