Some examples of forms of high rank

TL;DR

This paper presents new examples of forms with higher Waring rank in three and four variables, surpassing previous known ranks and the generic rank, thereby narrowing the possible maximum rank range.

Contribution

It introduces the first known forms in 3 and 4 variables with Waring ranks exceeding prior benchmarks and the generic rank, especially for degrees ≥ 5.

Findings

Forms in 3 variables with higher rank than monomials

Forms in 4 variables with rank greater than generic rank

First examples of forms with rank exceeding generic value for degrees ≥ 5

Abstract

We describe some forms with greater Waring rank than previous examples. In $3$ variables we give forms of odd degree with strictly greater rank than the ranks of monomials, the previously highest known rank. This narrows the possible range of values of the maximum Waring rank of forms in $3$ variables. In $4$ variables we give forms of odd degree with strictly greater than generic rank. In degrees greater than or equal to $5$ these are the first examples showing that there exist forms with Waring rank strictly greater than the generic value.