144 research outputs found
A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time
We describe a deterministic algorithm that computes an approximate root of n
complex polynomial equations in n unknowns in average polynomial time with
respect to the size of the input, in the Blum-Shub-Smale model with square
root. It rests upon a derandomization of an algorithm of Beltr\'an and Pardo
and gives a deterministic affirmative answer to Smale's 17th problem. The main
idea is to make use of the randomness contained in the input itself
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
On the number of minima of a random polynomial
We give an upper bound in O(d ^((n+1)/2)) for the number of critical points
of a normal random polynomial with degree d and at most n variables. Using the
large deviation principle for the spectral value of large random matrices we
obtain the bound
O(exp(-beta n^2 + (n/2) log (d-1))) (beta is a positive constant independent
on n and d) for the number of minima of such a polynomial. This proves that
most normal random polynomials of fixed degree have only saddle points.
Finally, we give a closed form expression for the number of maxima (resp.
minima) of a random univariate polynomial, in terms of hypergeometric
functions.Comment: 22 pages. We learned since the first version that the probability
that a matrix in GOE(n) is positive definite is known. This follows from the
theory of large deviations (reference in the paper). Therefore, we can now
state a precise upper bound (Theorem 2) for the number of minima of a random
polynomial, instead of a bound depending on that probabilit
Random systems of polynomial equations. The expected number of roots under smooth analysis
We consider random systems of equations over the reals, with equations
and unknowns , , , where the
's are non-random polynomials having degrees 's (the "signal") and
the 's (the "noise") are independent real-valued Gaussian centered random
polynomial fields defined on , with a probability law satisfying
some invariance properties. For each , and have degree .
The problem is the behavior of the number of roots for large . We prove that
under specified conditions on the relation signal over noise, which imply that
in a certain sense this relation is neither too large nor too small, it follows
that the quotient between the expected value of the number of roots of the
perturbed system and the expected value corresponding to the centered system
(i.e., identically zero for all ), tends to zero geometrically
fast as tends to infinity. In particular, this means that the behavior of
this expected value is governed by the noise part.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ149 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Average volume, curvatures, and Euler characteristic of random real algebraic varieties
We determine the expected curvature polynomial of random real projective
varieties given as the zero set of independent random polynomials with Gaussian
distribution, whose distribution is invariant under the action of the
orthogonal group. In particular, the expected Euler characteristic of such
random real projective varieties is found. This considerably extends previously
known results on the number of roots, the volume, and the Euler characteristic
of the solution set of random polynomial equationsComment: 38 pages. Version 2: corrected typos, changed some notation, rewrote
proof of Theorem 5.
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