Algebraicity of the "outer" boundary of the Mandelbrot set Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as 
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu \in M\rbrace.
$$

Question: Is $t_{\lambda,\mu}$ an algebraic number?

(If it's any easier - I'm also interested in the case when $\lambda$ and $\mu$ are both Gaussian rationals.)
 A: Clearly every piecewise real algebraic curve has this property. In general, I think you should expect that any curve with a simple, natural description, except for piecewise algebraic curves, does not have this property - it's simply too much of a conspiracy.
I will briefly review how to engineer such a conspiracy. Simply count the set of pairs of algebraic numbers, and step through them one by one. Start with any piecewise algebraic curve, and at each step, change it, either by increasing the degree or increasing the number of algebraic components, in a way that fixes the $t_{\lambda,\mu}$s of pairs $\lambda,\mu$ already counted. By the unbounded degrees of freedom available in such curves, this is clearly possible. One can also choose to do this while making only an incredibly small change to the overall shape of the curve, so that the sequence of curves converges uniformly to a limit, which cannot be algebraic as the complexity went to $\infty$, but which satisfies the requisite property.
But I see no reason to suspect the Mandelbrot set was created in this way. Note that the critical point is related not to one polynomial, but to an infinite sequence of polynomials, so even if it has a simple description, it is not so clear why that description should be an algebraic function. There are many other sorts of simple descriptions!
A: Of course, the answer depends on $\lambda$ and $\mu$. For example, if $\lambda=0$ and $\mu$ is $1$
or $-1$, then $t$ is rational.
On the other hand, even if $M$ is the unit disc, rather than Mandelbrot set, your number $t$ will
be transcendental for some $\mu$ and $\lambda$.
So probably you want to rephrase your question somehow. 
EDIT: Now I see that $\lambda$ and $\mu$ are algebraic numbers. (Sorry I did not see it when I read the
question first time). Then the following results may be relevant.


*

*The boundary of the Mandelbrot set is not a semi-algebraic set
(I believe this is in  MR0974426  L. Blum, M. Shub, S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 1, 1–46.

*MR2466298  M. Braverman, M. Yampolsky, Computability of Julia sets. Algorithms and Computation in Mathematics, 23. Springer-Verlag, Berlin, 2009.
