A closed proof of existence of solutions to third Painlev\'e' equation (Painlev\'e' transcendents) vanishing at arbitrary given point except zero is presented. The structure of the set of Painlev\'e' transcendents with common parameters vanishing at a given point is described. The approximate explicit representation of a Painlev\'e' transcendent in vicinity of its non-zero root is given.
The triple-zero Painlevé I transcendent
Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$
Properties of Quasi-Assouad dimension
![QR code for arXiv:2412.00170 [math.CA]](http://oss.arxitics.com/images/favicon.svg)