The asymptotic leading term for maximum rank of ternary forms of a given degree

Let rmax(n,d)rmax(n,d) be the maximum Waring rank for the set of all   homogeneous polynomials of degree d>0d>0 in n   indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when n,d≥3n,d≥3, the value of rmax(n,d)rmax(n,d) is known only for (n,d)=(3,3),(3,4),(3,5),(4,3)(n,d)=(3,3),(3,4),(3,5),(4,3). We prove that rmax(3,d)=d2/4+O(d)rmax(3,d)=d2/4+O(d) as a consequence of the upper bound rmax(3,d)≤⌊(d2+6d+1)/4⌋rmax(3,d)≤⌊(d2+6d+1)/4⌋.