9,194 research outputs found

    Polynomial systems with few real zeroes

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    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 Zn{\Z}^n, this bound is 2n+12n+1, while the Khovanskii bound is exponential in n2n^2. The bound 2n+12n+1 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

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    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

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    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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