Asymptotic Contractivity of Bohnenblust-Hille Constants with Bounded Monomial Support

We study the optimal Bohnenblust-Hille constants $K_{m,M}$ for complex $m$-homogeneous polynomials in any number of variables whose monomials with nonzero coefficient involve at most $M$ variables. For every fixed $M$, we prove that these constants satisfy $$K_{m,M}\le A_M^{M\mathrm{/}m}\cdot m^{(M^2-1)\mathrm{/}(2m)},\text{for }m\ge M,$$ where $A_M\ge 1$ depends only on $M$. In particular, $$K_{m,M}\to 1\text{as }m\to \mathrm{\infty },$$ and so the corresponding Bohnenblust-Hille constants are asymptotically contractive. The proof exploits the homogeneous structure through a decomposition according to exact monomial-support levels, partitions of the set of variables, multilinear Bohnenblust-Hille estimates, and interpolation with Parseval’s identity.