On generic and maximal k-ranks of binary forms

TL;DR

This paper investigates conjectures about the decomposition of homogeneous polynomials into sums of powers, focusing on the generic and maximal k-ranks of binary forms, and provides partial resolutions for specific cases.

Contribution

It proves the conjecture for the generic k-rank in two-variable cases and the maximal k-rank for binary forms of third powers of quadratics.

Findings

Confirmed the conjecture for two-variable forms.

Resolved the maximal k-rank for third powers of quadratic binary forms.

Advanced understanding of polynomial decompositions in specific cases.

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.