On generic and maximal k-ranks of binary forms

Samuel Lundqvist; Alessandro Oneto; Bruce Reznick; Boris Shapiro

doi:10.1016/j.jpaa.2018.08.015

On generic and maximal k-ranks of binary forms

Samuel Lundqvist, Alessandro Oneto, Bruce Reznick, Boris Shapiro

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k in any number of variables. The second one (by the fourth author) deals with the maximal k-rank of binary forms. We settle the first conjecture in the cases of two variables and the second in the first non-trivial case of the 3-rd powers of quadratic binary forms.

Original language	English (US)
Pages (from-to)	2062-2079
Number of pages	18
Journal	Journal of Pure and Applied Algebra
Volume	223
Issue number	5
DOIs	https://doi.org/10.1016/j.jpaa.2018.08.015
State	Published - May 2019

ASJC Scopus subject areas

Algebra and Number Theory

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title = "On generic and maximal k-ranks of binary forms",

abstract = "In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k in any number of variables. The second one (by the fourth author) deals with the maximal k-rank of binary forms. We settle the first conjecture in the cases of two variables and the second in the first non-trivial case of the 3-rd powers of quadratic binary forms.",

author = "Samuel Lundqvist and Alessandro Oneto and Bruce Reznick and Boris Shapiro",

note = "Funding Information: Acknowledgments We want to thank the participants of the problem-solving seminar in commutative algebra at Stockholm University and especially Ralf Fr?berg for creating a nice research atmosphere. It is a pleasure to acknowledge the importance of our communication with Giorgio Ottaviani for the present study. The third author was supported in part by Simons Foundation Grant 280987. Finally, the fourth author is sincerely grateful to the University of Illinois, Urbana?Champaign for the hospitality in summer 2015 when a part of this project was carried out.",

year = "2019",

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doi = "10.1016/j.jpaa.2018.08.015",

language = "English (US)",

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AU - Lundqvist, Samuel

AU - Oneto, Alessandro

AU - Reznick, Bruce

AU - Shapiro, Boris

N1 - Funding Information: Acknowledgments We want to thank the participants of the problem-solving seminar in commutative algebra at Stockholm University and especially Ralf Fr?berg for creating a nice research atmosphere. It is a pleasure to acknowledge the importance of our communication with Giorgio Ottaviani for the present study. The third author was supported in part by Simons Foundation Grant 280987. Finally, the fourth author is sincerely grateful to the University of Illinois, Urbana?Champaign for the hospitality in summer 2015 when a part of this project was carried out.

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