Complexity of Bezout's Theorem V: Polynomial Time

Complexity of Bezout's Theorem VII: Distance Estimates in the Condition Metric

We study geometric properties of the solution variety for the problem of approximating solutions of systems of polynomial equations. We prove that given two pairs (fi, ζi), i = 1, 2, there exists a short path joining them such that the complexity of following the path is bounded by the logarithm of the condition number of the problems.

Derived Algebraic Geometry V: Structured Spaces Introduction: Bezout's Theorem

On the Determinant of a Uniformly Distributed Complex Matrix

We derive the joint density for the singular values of a random complex matrix A uniformly distributed on kAk F = 1. This joint density allows us to obtain the conditional expectation of det(A H A) = j det Aj 2 given the smallest singular value. This result has been used by Shub and Smale in their analysis of the complexity of Bezout's Theorem.

Efficient implementation of low time complexity and pipelined bit-parallel polynomial basis multiplier over binary finite fields

This paper presents two efficient implementations of fast and pipelined bit-parallel polynomial basis multipliers over GF (2m) by irreducible pentanomials and trinomials. The architecture of the first multiplier is based on a parallel and independent computation of powers of the polynomial variable. In the second structure only even powers of the polynomial variable are used. The par...

The polynomial and linear time hierarchies in V0

We show that the bounded arithmetic theory V does not prove that the polynomial time hierarchy collapses to the linear time hierarchy (without parameters). The result follows from a lower bound for bounded depth circuits computing prefix parity, where the circuits are allowed some auxiliary input; we derive this from a theorem of Ajtai.