Maximum Waring Ranks of Monomials and Sums of Coprime Monomials

Abstract

Abstract. We show that monomials and sums of pairwise coprime monomials in four or more variables have Waring rank less than the generic rank, with a short list of exceptions. We asymptotically compare their ranks with the generic rank.

Seeking for the Maximum Symmetric Rank

“…For a detailed discussion on maximum rank of monomials, we refer the reader to [20]. Let us now outline how the rank of monomials has been bounded from below, and how that technique has been enhanced by Buczyński and Teitler, to exceed the previously known lower bounds on maximum ranks.…”

Some examples of forms of high rank

“…For cubic surfaces r max (4, 3) = 7 while r gen (4, 3) = 5, see [24, §97]. (See [16] for the form F = x 1 x 2 2 + x 3 x 2 4 of degree d = 3 in n = 4 variables which has rank 6.) To our knowledge, the maximum Waring rank is not known up to now for any other values of (n, d).…”