Waring rank of binary forms, harmonic cross-ratio and golden ratio

Abstract

We discuss the Waring rank of binary forms of degree 4 and 5, without multiple factors, and point out unexpected relations to the harmonic cross-ratio, j-invariants and the golden ratio. These computations of ranks for binary forms are used to show that the combinatorics of a line arrangement in the complex projective plane does not determine the Waring rank of the defining equation even in very simple situations.

arXiv commentv.3: We relate our results to results in the paper L. Brustenga i Moncus\' i, S. K. Masuti, On the Waring rank of binary forms: the binomial formula and a dihedral cover of rank two forms, arXiv:1901.08320