Transcendence of polynomial canonical heights

There are two fundamental problems motivated by Silverman’s conversations over the years concerning nature of exact values canonical heights $$f(z)\in \bar{\mathbb {Q}}(z)$$ with $$d:=\deg (f)\ge 2$$ . The first problem is conjecture that $$\hat{h}_f(a)$$ either 0 or transcendental for every $$a\in \mathbb {P}^1(\bar{\mathbb {Q}})$$ ; this holds when f linearly conjugate to $$z^d$$ $$\pm C_d(z)$$ where $$C_d(z)$$ Chebyshev polynomial degree d since $$\hat{H}_f(a)$$ algebraic a. Other than this, very little known: example, it not known if there exists even one rational number a such irrational $$f(z)=z^2+\displaystyle \frac{1}{2}$$ second asks characterization all pairs (f, a) algebraic. In paper, we solve and obtain significant progress toward in case dynamics. These consequences our main result numbers can be expressed as multiplicative combination Böttcher coordinates. proof uses certain auxiliary powerful Medvedev–Scanlon classification preperiodic subvarieties split maps.