Sharp asymptotics of the homogeneous quotient Hilbert–Schmidt norm

Determine the sharp asymptotics of the supremum of the squared Hilbert–Schmidt norms of compressed cross-commutators over homogeneous degree-$D$ polynomial quotient modules, including whether the normalized supremum converges to one and whether powers of a single toral linear factor are asymptotically extremal.

Background

For a homogeneous polynomial generator of degree $D$, the paper proves an upper bound of $1+\log\binom{2D}{D}$ for the squared Hilbert–Schmidt norm of the quotient cross-commutator. The family $p_N=(z-w)^N$ gives a lower bound of order $D$ and shows that no degree-independent bound exists.

The exact leading constant, extremizing generators, and stability or equality conditions remain unresolved. The problem asks whether the lower-bound family is asymptotically optimal and how the relevant Verblunsky coefficients and endpoint weights characterize extremality.