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  • 1
    Electronic Resource
    Electronic Resource
    The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings (2000)
    Springer
    Journal of the European Mathematical Society 2 (2000), S. 179-198 
    Details
    ISSN: 1435-9863
    Keywords: Mathematics Subject Classification (1991): 52B11, 52B20, 14M25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum ?1+...+? r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding ?(?1,...,? r ). In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s100970050003
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Polyhedral end games for polynomial continuation (1998)
    Springer
    Numerical algorithms 18 (1998), S. 91-108 
    Details
    ISSN: 1572-9265
    Keywords: homotopy continuation ; polynomial systems ; Newton polytopes ; Bernshtein bound ; cycle number ; 14Q99 ; 52A39 ; 52B20 ; 65H10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Bernshtein’s theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (ℂ*)n, with ℂ* = ℂ\{0}. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (ℂ*)n. In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with approximations of its solutions. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019163811284
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    Paper (German National Licenses)
    Fulltext
  • 3
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    Unknown
    A family of sparse polynomial systems arising in chemical reaction systems (1999)
    Details
    Publication Date: 2014-02-26
    Description: A class of sparse polynomial systems is investigated which is defined by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace is of particular interest. We show that the simplest cases are equivalent to binomial systems while in general the solution structure is highly determined by the properties of the two graphs. First we recall results by Feinberg and give rigorous proofs. Secondly, we explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. The results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 4
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    The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings (1999)
    Details
    Publication Date: 2014-11-10
    Description: In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of \emph{coherent} mixed subdivisions of a Minkowski sum $\mathcal{A}_1+\cdots+\mathcal{A}_r$ of point configurations and of \emph{coherent} polyhedral subdivisions of the associated Cayley embedding $\mathcal{C}(\mathcal{A}_1,\dots,\mathcal{A}_r)$. In this paper we extend this correspondence in a natural way to cover also \emph{non-coherent} subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    OPUS
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