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Cayley Factorization and the Area Principle

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Abstract

In Sturmfels and Whiteley (J Symb Comput 11(5):439–453, 1991), it is proven that any multihomogenous bracket polynomial with integer coefficients can be interpreted by a ruler construction when introducing simple non-degeneracy conditions. This problem is called (generalized) Cayley factorization. This allows for geometrically interpreting many projective invariant properties. We reprove the above statement in rank 3 giving a better bound on the size of the non-degeneracy conditions. The constant factor in the bound is essentially reduced from 105 to 9. The algorithm described is concise enough to be implemented on a computer. With this algorithm, interpreting a common condition for six points to lie on a common conic is very close to Pascal’s construction (see Apel, in: The geometry of brackets and the area principle, Dissertation, TU München, 2014).

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Notes

  1. For evaluating the formula given in the case \(m\ge 2\), use the second version of (1) within the equivalent formula .

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  1. Zentrum Mathematik, Lehrstuhl für Geometrie und Visualisierung (M10), Technische Universität München, Boltzmannstr. 3, 85748, Garching bei München, Germany

    Susanne Apel

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Apel, S. Cayley Factorization and the Area Principle. Discrete Comput Geom 55, 203–227 (2016). https://doi.org/10.1007/s00454-015-9738-2

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  • DOI: https://doi.org/10.1007/s00454-015-9738-2

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