Yang’s monotonicity conjecture

Yang’s conjecture concerns monotonicity of numerical invariants associated with submodules of the Hardy space on the bidisc. The available results settle important power-type families, not all finitely generated polynomial submodules.

August 2026 family results

Bingyang Hu and Kunyu Guo report the result for the family $[(z-w)^k]$, extending the previously known cases $k=1,2$ to every $k\ge 1$. A separate August $13$, $2026$ preprint proves, for $M_k=[z^k-w^k]$ and $k\ge 2$, the block formula $\Sigma_j(M_k)=\Sigma_{\lceil j/k\rceil}([z-w])$ for $j\ge 1$, yielding nonincreasing sequences with strict decreases between blocks. Both results are explicitly limited to their stated families.

Current status (as of August 2026): Monotonicity is reported for the families $[(z-w)^k]$ and $[z^k-w^k]$, while Yang’s conjecture for general submodules remains open.