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Proceedings of the American Mathematical Society Series B

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On defectivity of families of full-dimensional point configurations


Authors: Christopher Borger and Benjamin Nill
Journal: Proc. Amer. Math. Soc. Ser. B 7 (2020), 43-51
MSC (2010): Primary 14M25, 52B20; Secondary 52A39, 13P15
DOI: https://doi.org/10.1090/bproc/46
Published electronically: May 15, 2020
MathSciNet review: 4098590
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Abstract: The mixed discriminant of a family of point configurations can be considered as a generalization of the $A$-discriminant of one Laurent polynomial to a family of Laurent polynomials. Generalizing the concept of defectivity, a family of point configurations is called defective if the mixed discriminant is trivial. Using a recent criterion by Furukawa and Ito we give a necessary condition for defectivity of a family in the case that all point configurations are full-dimensional. This implies the conjecture by Cattani, Cueto, Dickenstein, Di Rocco, and Sturmfels that a family of $n$ full-dimensional configurations in ${\mathbb {Z}}^n$ is defective if and only if the mixed volume of the convex hulls of its elements is $1$.


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Additional Information

Christopher Borger
Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
MR Author ID: 1353748
ORCID: 0000-0002-9735-394X
Email: christopher.borger@ovgu.de

Benjamin Nill
Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
MR Author ID: 754204
Email: benjamin.nill@ovgu.de

Keywords: Mixed discriminant, $A$-discriminant, defectivity, Cayley polytopes, lattice polytopes
Received by editor(s): October 22, 2019
Received by editor(s) in revised form: March 9, 2020
Published electronically: May 15, 2020
Additional Notes: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.
The second author is an affiliated researcher with Stockholm University and was partially supported by the Vetenskapsrådet grant NT:2014-3991.
Communicated by: Patricia Hersh
Article copyright: © Copyright 2020 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)