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Mathematics of Computation

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A numerical toolkit for multiprojective varieties


Authors: Jonathan D. Hauenstein, Anton Leykin, Jose Israel Rodriguez and Frank Sottile
Journal: Math. Comp. 90 (2021), 413-440
MSC (2010): Primary 65H10
DOI: https://doi.org/10.1090/mcom/3566
Published electronically: October 2, 2020
MathSciNet review: 4166467
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Abstract: A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections and to use this toolkit in an algorithm for numerical irreducible decomposition of multiprojective varieties. The toolkit and decomposition algorithm are illustrated throughout in a series of examples.


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Additional Information

Jonathan D. Hauenstein
Affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556
Email: hauenstein@nd.edu

Anton Leykin
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
Email: leykin@math.gatech.edu

Jose Israel Rodriguez
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: Jose@math.wisc.edu

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: sottile@math.tamu.edu

DOI: https://doi.org/10.1090/mcom/3566
Keywords: Numerical algebraic geometry, multiprojective variety
Received by editor(s): August 6, 2019
Received by editor(s) in revised form: April 23, 2020
Published electronically: October 2, 2020
Additional Notes: The research of the first author was supported in part by NSF grant CCF-1812746.
The research of the second author was supported in part by NSF grant DMS-1151297.
The research of the third author was supported in part by NSF grant DMS-1402545.
The research of the fourth author was supported in part by NSF grant DMS-1501370.
Article copyright: © Copyright 2020 American Mathematical Society