52 research outputs found
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
Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces
We prove that any pair of bivariate trinomials has at most 5 isolated roots
in the positive quadrant. The best previous upper bounds independent of the
polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate
roots) via a famous general result of Khovanski. Our bound is sharp, allows
real exponents, allows degeneracies, and extends to certain systems of
n-variate fewnomials, giving improvements over earlier bounds by a factor
exponential in the number of monomials. We also derive analogous sharpened
bounds on the number of connected components of the real zero set of a single
n-variate m-nomial.Comment: 27 pages, 2 figures. Extensive revision of math.CO/0008069. To appear
in Discrete and Computational Geometry. Technique from main theorem (Theorem
1) now pushed as far as it will go. In particular, Theorem 1 now covers
certain fewnomial systems of type (n+1,...,n+1,m) and certain non-sparse
fewnomial systems. Also, a new result on counting non-compact connected
components of fewnomial hypersurfaces (Theorem 3) has been adde
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
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