32,307 research outputs found
An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions
We prove an asymptotically tight bound (asymptotic with respect to the number
of polynomials for fixed degrees and number of variables) on the number of
semi-algebraically connected components of the realizations of all realizable
sign conditions of a family of real polynomials. More precisely, we prove that
the number of semi-algebraically connected components of the realizations of
all realizable sign conditions of a family of polynomials in
whose degrees are at most is bounded by This improves the best upper bound known
previously which was The new
bound matches asymptotically the lower bound obtained for families of
polynomials each of which is a product of generic polynomials of degree one.Comment: 19 pages. Bibliography has been updated and a few more references
have been added. This is the final version of this paper which will appear in
Combinatoric
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
We give new positive and negative results (some conditional) on speeding up
computational algebraic geometry over the reals: (1) A new and sharper upper
bound on the number of connected components of a semialgebraic set. Our bound
is novel in that it is stated in terms of the volumes of certain polytopes and,
for a large class of inputs, beats the best previous bounds by a factor
exponential in the number of variables. (2) A new algorithm for approximating
the real roots of certain sparse polynomial systems. Two features of our
algorithm are (a) arithmetic complexity polylogarithmic in the degree of the
underlying complex variety (as opposed to the super-linear dependence in
earlier algorithms) and (b) a simple and efficient generalization to certain
univariate exponential sums. (3) Detecting whether a real algebraic surface
(given as the common zero set of some input straight-line programs) is not
smooth can be done in polynomial time within the classical Turing model (resp.
BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily
from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of
Complexity. More typos are corrected, and a new section is added where the
bounds here are compared to an earlier result of Benedetti, Loeser, and
Risler. The LaTeX source needs the ajour.cls macro file to compil
An improved bound on the number of point-surface incidences in three dimensions
We show that points and smooth algebraic surfaces of bounded degree
in satisfying suitable nondegeneracy conditions can have at most
incidences, provided that any
collection of points have at most O(1) surfaces passing through all of
them, for some . In the case where the surfaces are spheres and no
three spheres meet in a common circle, this implies there are point-sphere incidences. This is a slight improvement over the previous
bound of for an (explicit) very
slowly growing function. We obtain this bound by using the discrete polynomial
ham sandwich theorem to cut into open cells adapted to the set
of points, and within each cell of the decomposition we apply a Turan-type
theorem to obtain crude control on the number of point-surface incidences. We
then perform a second polynomial ham sandwich decomposition on the irreducible
components of the variety defined by the first decomposition. As an
application, we obtain a new bound on the maximum number of unit distances
amongst points in .Comment: 17 pages, revised based on referee comment
New Complexity Bounds for Certain Real Fewnomial Zero Sets
Consider real bivariate polynomials f and g, respectively having 3 and m
monomial terms. We prove that for all m>=3, there are systems of the form (f,g)
having exactly 2m-1 roots in the positive quadrant. Even examples with m=4
having 7 positive roots were unknown before this paper, so we detail an
explicit example of this form. We also present an O(n^{11}) upper bound for the
number of diffeotopy types of the real zero set of an n-variate polynomial with
n+4 monomial terms.Comment: 8 pages, no figures. Extended abstract accepted and presented at MEGA
(Effective Methods in Algebraic Geometry) 200
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