For strictly convex polynomials \(b:\mathbb R^n\longrightarrow\mathbb R\) of combined degree \((m_1,\dots,m_n)\in\mathbb N^n\) (i.e., of the form
\[ p(x)=\sum_{\alpha}c_{\alpha}x^{\alpha}, \] where the exponents of its pure terms of highest order are \(2m_1,\dots,2m_n\), respectively, and \[ \sum_{j=1}^n\frac{\alpha_j}{2m_j}\leq1 \] with equality if and only if there is a \(j\) such that \(\alpha_j=2m_j\)), define \[ \Omega_b:=\big\{(z_1,\dots,z_{n+1})\in\mathbb C^{n+1}:\operatorname{Im} z_{n+1}>b(\operatorname{Re} z_1,\dots,\operatorname{Re} z_n)\big\}. \] We use the following identification of the boundary of \(\Omega_b\) \[ \mathbb C^n\times\mathbb R\ni (z,t)\longmapsto(z,t+ib(\operatorname{Re} z_1,\dots,\operatorname{Re} z_n))\in\partial\Omega_b. \] Finally, let \(S\) denote the Szegő kernel of \(\Omega_b\).
The main result of the paper under review is the following estimate. Let \((x,y,t), (x',y',t')\in\partial\Omega_b\) and define \[ \tilde b(v)=b\Big(v+\frac{x+x'}{2}\Big)-\nabla b\Big(\frac{x+x'}{2}\Big)v-b\Big(\frac{x+x'}{2}\Big),\quad \delta(x,x')=b(x)+b(x')-2b\Big(\frac{x+x'}{2}\Big), \] and \[ w=t'-t+\nabla b\Big(\frac{x+x'}{2}\Big)(y'-y). \] Then \[ |S((x,y,t),(x',y',t'))|\leq c\Big(\sqrt{\delta^2+\tilde b(y-y')^2+w^2}\Big|\Big\{v:\tilde b(v)<\sqrt{\delta^2+\tilde b(y-y')^2+w^2}\Big\}\Big|^2\Big)^{-1}, \] where the constant \(c\) depends only on the combined degree \((m_1,\dots,m_n)\) and the dimension of the space.