Start with any polynomial of degree $n$ with complex coefficients, e.g., $$z^3+z^2+2 z+3 \;.$$ Find its $n$ roots, and list them in order of their modulus: $$-1.28, (0.14\pm 1.53 i)$$ Now form a new polynomial with leading coefficient $1$ and the other coefficients the $n$ roots: $$ z^3 -1.28\, z^2 +(0.14 -1.53 i)\, z +(0.14 +1.53 i) \;. $$ Here I used the sorting convention that $(a-b i) < (a + b i)$. A bit more formally, let $P$ be $$ P \;:\; z^n + a_{n-1}z^{n-1} + \cdots + a_0 \;, $$ with roots $$r_{n-1}, \ldots ,r_0 \;,$$ sorted so that $|r_i| \le |r_{i-1}|$. Define $P_1$ as $$ P_1 \;:\; z^n + r_{n-1}z^{n-1} + \cdots + r_0 \;. $$ My question is:

. Which are the polynomials whose roots are its coefficients, in the sense that $r_i=a_i$, $i=0,\ldots,n-1$, i.e., $P_1 = P$?Q1

Here is one fixed point: $$ z^3 -(0.782599 +0.521714 i)\, z^2 +(0.884646 -0.589743 i)\, z +(0.680552 +1.63317 i) $$

When this process is iterated, it appears to fall into cycles, not always of length $1$.

. Does iteration of this process always fall into a cycle?Q2

(**Added.** *3Sep15*.) The question suggested by Igor Rivin is also interesting,
especially in light of Richard Stanley's remark concerning real polynomials:

. Which are the polynomials whose roots are its coefficients, in any order? (This is an unordered version of Q1.)Q3

(**Added.** *4Sep15*.)
Concerning Q2, which I realize is less interesting than the other questions, here is
an example of a cycle of length $11$. I plot the magnitude of the vector
of $3$ roots versus iteration #, starting from a random cubic polynomial. The last
vector of roots is
$$
(-1.52+0.88 i,-0.48-0.90 i,1.00 +0.02 i)
$$
with magnitude $2.27$: