ABSTRACT: Let f be a sparse random polynomial system. This means that the support (list of possibly non-zero coefficients) of each coordinate polynomial fi is fixed, and each fi is an independent random variable, with Gaussian probability distribution (arbitrary covariance).
We express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form. When U = (C*)n, the classical mixed volume is recovered.
A bound is also given for the probability that the condition number of f on the region U is larger than 1/ε. This bound depends on the integral of the mixed volume form over U, and on a certain intrinsic invariant of U as a subset of a toric manifold.
Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given.
The connection between zeros of sparse random polynomial systems, Kähler geometry, and mechanics (momentum maps) is discussed.
This paper is dedicated to Steve Smale on his 70th birthday.
ABSTRACT: Kronecker's theorem states that if 1, θ1, ... , θn are real algebraic numbers, linearly independent over Q, and if α is in N, then for any ε > 0 there are q in Z and p in Zn such that |q θi - αi - pi | < ε.
Here, a bound on q is given in terms of the dimension n, of the precision ε, of the degree of the θi's and of their height.
A possible connection to the square-root sum problem is discussed.
WARNING: I withdrew this article after I found out that earlier results by Erdös and Turán, using discrepancy, led to sharper results.
ABSTRACT: An elementary proof of the invariance of the Kostlan norm of the space of all polynomial systems, under unitary transformation.
ABSTRACT: A worst case bound for the condition number of a generic system of polynomial equations with integer coefficients is given. For fixed degree and number of equations, the condition number is (non-uniformly, generically) pseudo-polynomial in the input size.
SEE "Lower bounds for some decision problems over C"
ABSTRACT: The number of steps of homotopy algorithms for solving systems of polynomials is usually bounded by the condition number of the homotopy path. A generic bound on the condition number of homotopy path between systems with integer coefficients will be given.
Chapter 2 became the paper "On generalized Newton algorithms : Quadratic convergence, path-following and error analysis". Chapter 4 is partially superseded by "Lower bounds for some decision problems over C"
