• Publications
  • Influence
Hilbert's Nullstellensatz Is in the Polynomial Hierarchy
TLDR
A new lower bound for unsatisfiable systems is included, and remarks on the Arthur-Merlin class are included, on the problem of deciding whether a system of polynomial equations in several complex variables has a solution.
A weak version of the Blum, Shub and Smale model
  • P. Koiran
  • Computer Science, Mathematics
    Proceedings of IEEE 34th Annual Foundations of…
  • 3 November 1993
TLDR
A weak version of the Blum-Shub-Smale model of computation over the real numbers is proposed, where only a "moderate" usage of multiplications and divisions is allowed, and the class of languages recognizable in polynomial time is shown to be the complexity class P/poly.
Arithmetic circuits: The chasm at depth four gets wider
  • P. Koiran
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 24 June 2010
Shallow circuits with high-powered inputs
  • P. Koiran
  • Computer Science, Mathematics
    ICS
  • 27 April 2010
TLDR
It is shown that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent, and the intriguing possibility that tools from real analysis might be brought to bear on a longstanding open problem.
Approximating the volume of definable sets
  • P. Koiran
  • Computer Science, Mathematics
    Proceedings of IEEE 36th Annual Foundations of…
  • 23 October 1995
TLDR
An upper bound on the precision that should be used in a Monte-Carlo integration method is given and an application to a problem of structural complexity in the Blum-Shub-Smale model of computation over the reals is described.
Computability with Low-Dimensional Dynamical Systems
Neural Networks with Quadratic VC Dimension
TLDR
It is shown that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weightsw, which settles a long-standing open question.
On the complexity of partial derivatives
TLDR
The "trace method", recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices, is analyzed and it is shown that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method the authors derive closed-form lower bounds.
Valiant’s model and the cost of computing integers
  • P. Koiran
  • Mathematics, Computer Science
    computational complexity
  • 1 February 2005
TLDR
It is shown that a proof of this conjecture for the first sequence would imply a superpolynomial lower bound for the arithmetic circuit size of the permanent polynomial or P ≠ PSPACE.
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