On defectivity of families of full-dimensional point configurations

Borger, Christopher; Nill, Benjamin

doi:10.1090/bproc/46

On defectivity of families of full-dimensional point configurations
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by Christopher Borger and Benjamin Nill HTML | PDF

Proc. Amer. Math. Soc. Ser. B 7 (2020), 43-51

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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