Unlikely intersection problems for multi-parameter families of polynomial maps


Liang-Chung Hsia ( National Taiwan Normal University )
2017-02-24  13:30 - 14:30
Room 103, Mathematics Research Center Building (ori. New Math. Bldg.)

Let $d>m>1$ be integers, let $c_1,\dots, c_{\ell}$ be distinct complex numbers, and let ${\mathbf f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m$ be an $m$-parameter family of polynomials. We're interested in the set $S(c_1,\ldots, c_\ell)$ of $m$-tuples of parameters $(t_1,\dots, t_m)\in {\mathbb C}^m$ with the property that each $c_i$ (for $i=1,\dots, m+1$) is preperiodic under the action of the corresponding polynomial ${\mathbf f}(z)$. In particular, we will show that if $\ell \ge m+1$ then $S(c_1, \ldots, c_{\ell})$ is contained in finitely many hypersurfaces of the parameter space ${\mathbf A}^m$. This is a joint work with D. Ghioca and K. D. Nguyen.