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:
Q1. Which are the polynomials whose roots are its coefficients,
in the sense that $r_i=a_i$, $i=0,\ldots,n-1$, i.e., $P_1 = P$?
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 $1$.
Q2. Does iteration of this process always fall into a cycle?
(Added. 3Sep15.) The question suggested by Igor Rivin is also interesting,
especially in light of Richard Stanley's remark concerning real polynomials:
Q3. Which are the polynomials whose roots are its coefficients, in any order? (This is an unordered version of Q1.)
(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 $2.27$:
Best Answer
It is known (though I don't have a reference except for the statement at http://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/assignments/roots.pdf, #2) that the only real polynomials with nonzero constant term whose roots equal their coefficients are $x^2+x-2$, $x^3+x^2-x-1$, and (approximately) $$ x^3 + .56519772x^2 - 1.76929234x + .63889690 $$ $$ x^4 + x^3 - 1.7548782x^2 - .5698401x + .3247183. $$ Update. I found a reference: P. Stein, On polynomial equations with coefficients equal to their roots, Amer. Math. Monthly 73 (1966), 272-274.