9,194 research outputs found
Polynomial systems with few real zeroes
We study some systems of polynomials whose support lies in the convex hull of
a circuit, giving a sharp upper bound for their numbers of real solutions. This
upper bound is non-trivial in that it is smaller than either the Kouchnirenko
or the Khovanskii bounds for these systems. When the support is exactly a
circuit whose affine span is , this bound is , while the
Khovanskii bound is exponential in . The bound can be attained only
for non-degenerate circuits. Our methods involve a mixture of combinatorics,
geometry, and arithmetic.Comment: 23 pages, 1 .eps figure. Revised Introductio
Polynomial systems supported on circuits and dessins d'enfants
We study polynomial systems whose equations have as common support a set C of
n+2 points in Z^n called a circuit. We find a bound on the number of real
solutions to such systems which depends on n, the dimension of the affine span
of the minimal affinely dependent subset of C, and the "rank modulo 2" of C. We
prove that this bound is sharp by drawing so-called dessins d'enfant on the
Riemann sphere. We also obtain that the maximal number of solutions with
positive coordinates to systems supported on circuits in Z^n is n+1, which is
very small comparatively to the bound given by the Khovanskii fewnomial
theorem.Comment: 19 pages, 5 figures, Section 3.1 revised, minor changes in other
section
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
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