## On defectivity of families of full-dimensional point configurations

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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 $1$.## References

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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
- 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
- © Copyright 2020 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- 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
- MathSciNet review: 4098590