Shanzhong Sun

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood.

We address the following question: let $F:(\mathbb {R}^2,0)\to(\mathbb {R}^2,0)$ be an analytic local diffeomorphism defined in the neighborhood of the nonresonant elliptic fixed point 0 and let $\Phi$ be a formal conjugacy to a normal form $N$. Supposing $F$ leaves invariant the foliation by circles centered at $0$, what is the analytic nature of $\Phi$ and $N$?