### Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains

For a strongly pseudoconvex domain $D\subset {\u2102}^{n+1}$ defined by a real polynomial of degree ${k}_{0}$, we prove that the Lie group $\mathrm{Aut}\left(D\right)$ can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle $Y$ of $\partial D$, and that the sum of its Betti numbers is bounded by a certain constant ${C}_{n,{k}_{0}}$ depending only on $n$ and ${k}_{0}$. In case $D$ is simply connected, we further give an explicit but quite rough bound in terms of the dimension and the degree of the defining polynomial. Our approach is to adapt the Cartan-Chern-Moser...