On the Degree Growth of Birational Mappings in Higher Dimension

Let $f$ be a birational map of ${\mathbf{C}}^d$, and consider the degree complexity, or asymptotic degree growth rate $\delta(f)=\lim_{n\to\infty}({\rm deg}(f^n))^{1/n}$. We introduce a family of elementary maps, which have the form $f=L\circ J$, where $L$ is (invertible) linear, and $J(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_d)=(x_1^{-1},\mathrm{.}\mathrm{.}\mathrm{.},x_d^{-1})$. We develop a method of regularization and show how it can be used to compute $\delta$ for an elementary map.