We prove that the radii of convergence of the solutions of a$p$-adic differential equation $${\fancyscript{F}}$$ over an affinoid domain$X$of the Berkovich affine line are continuous functions on$X$that factorize through the retraction of $${X\to\Gamma}$$ of$X$onto a finite graph $${\Gamma\subseteq X}$$ . We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on$X$is controlled by a$finite$family of data.

No abstract uploaded!

It is shown that the punctual quotient scheme$Q$_{l}^{r}parametrizing all zero-dimensional quotients $$\mathcal{O}_{A^2 }^{ \oplus ^r } \to T$$ of length$l$and supported at some fixed point O∈$A$^{2}in the plane is irreducible.

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be “zipped” down to a strikingly small data set.

No abstract uploaded!