On the multiplicative complexity of polynomials

We study the multiplicative (nonscalar) complexity $\mu(M_{d,n})$ for the class $M_{d,n}$ of complex polynomials in $n$ variables of degree at most $d$. It is shown that the relation $\mu(M_{d,n}) \asymp n^{\lceil d/2\rceil}$ holds for any constant $d \ge 2$. The lower bound for odd values of $d$ in this relation is new. For the case of cubic polynomials we prove more accurate bounds $n^2/18 \lesssim \mu(M_{3,n}) \lesssim n^2/4$.