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Consider a family of polynomials $\mathcal{F}$. Let $p$ be a single complex point or a finite set of complex points inside the unit disk.

I am interested in what can we say about the distribution of

$$\{ f(p): f \in \mathcal{F} \},$$

where $\mathcal{F}$ is a family of polynomial such that all their roots are in the unit disk.

Let me make it more precise what I mean by the distribution. Let $M$ be a large number and we divide $[-1, 1]$ to $2M$ equal segment, we call this set $\mathcal{M}$. Let

$$\mathcal{R}= \{r_1+i r_2 \in \mathcal{M} + i\mathcal{M}: r^2_{1}+ r^2_{2}<1 \}.$$ Which is the set of lattice points inside the unite disk generated by $\{1/M, i/M\}$. Let

$$\mathcal{F} = \{f \text{ is a polynomial of degree } < X \text{ such that } \text{ if } f(r)=0 \text{ then } r \in \mathcal{R} \},$$

which is an "approximation" of family of polynomials with their roots inside the unit disk. The size of $\mathcal{F}$ is $\mathcal{R}^X.$

Assume that we $p$ is a fixed point inside the unite circle and define:

$$\mathcal{T} = \{f(p) : f \in \mathcal{F} \}.$$

Now define the density function as

$$\text{pdf}(x)= \frac{\#\{t \in T : |t|< x\}}{\#\{t \in T\}}.$$

Questions are:

-What does $\text{pdf}$ look like?

-Does it depend on the initial point $p$?

-What happens if we change the family of polynomials?

I wrote a code that suggest it may be normally distributed, however, computationally, it is had to go above degree $4$.

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