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

Abstract : In what follows, we pose two general conjectures about decompositions of homogeneous poly-nomials 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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Samuel Lundqvist, Alessandro Oneto, Bruce Reznick, Boris Shapiro. On generic and maximal $k$-ranks of binary forms. 2017. ⟨hal-01637944⟩

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