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Projective Geometry of Wachspress Coordinates
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Matematisk institutt, Universitetet i Oslo, PB 1053, Blindern, 0316, Oslo, Norway.
2019 (English)In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383Article in journal (Refereed) Published
Abstract [en]

We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

Place, publisher, year, edition, pages
2019.
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-268299DOI: 10.1007/s10208-019-09441-zScopus ID: 2-s2.0-85075169402OAI: oai:DiVA.org:kth-268299DiVA, id: diva2:1414482
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QC 20200313

Available from: 2020-03-13 Created: 2020-03-13 Last updated: 2020-04-07

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Publisher's full textScopushttps://link.springer.com/article/10.1007%2Fs10208-019-09441-z

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Kohn, Kathlén

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