M\"obius function of coordinate hyperplanes in complex ellipsoids

Abstract

For $p_1,...,p_n>0$, let $\mathbb E=\{z\in\mathbb C^n:\sum_{j=1}^n|z_j|^{2p_j}<1\}$ be a complex ellipsoid. We present effective formulas for the generalized M\"obius and Green functions $m_{\mathbb E}(A,\cdot)$, $g_{\mathbb E}(A,\cdot)$ in the case where $A:=\{z\in\mathbb E:z_1\cdot...\cdot z_k=0\}$ ($1\leq k\leq n$).