TY - JOUR

T1 - On generic and maximal k-ranks of binary forms

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.

PY - 2019/5

Y1 - 2019/5

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jpaa.2018.08.015

DO - 10.1016/j.jpaa.2018.08.015

M3 - Article

AN - SCOPUS:85052157255

VL - 223

SP - 2062

EP - 2079

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 5

ER -