113 research outputs found
Solving Degenerate Sparse Polynomial Systems Faster
Consider a system F of n polynomial equations in n unknowns, over an
algebraically closed field of arbitrary characteristic. We present a fast
method to find a point in every irreducible component of the zero set Z of F.
Our techniques allow us to sharpen and lower prior complexity bounds for this
problem by fully taking into account the monomial term structure. As a
corollary of our development we also obtain new explicit formulae for the exact
number of isolated roots of F and the intersection multiplicity of the
positive-dimensional part of Z. Finally, we present a combinatorial
construction of non-degenerate polynomial systems, with specified monomial term
structure and maximally many isolated roots, which may be of independent
interest.Comment: This is the final journal version of math.AG/9702222 (``Toric
Generalized Characteristic Polynomials''). This final version is a major
revision with several new theorems, examples, and references. The prior
results are also significantly improve
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
Mixed Volume Techniques for Embeddings of Laman Graphs
Determining the number of embeddings of Laman graph frameworks is an open
problem which corresponds to understanding the solutions of the resulting
systems of equations. In this paper we investigate the bounds which can be
obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is
to provide the methods to study the mixed volume of suitable systems of
polynomial equations obtained from the edge length constraints. While in most
cases the resulting bounds are weaker than the best known bounds on the number
of embeddings, for some classes of graphs the bounds are tight.Comment: Thorough revision of the first version. (13 pages, 4 figures
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