How to find area of a polygon built on the roots of a given polynomial? How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane?
For example, consider the equation:
$$2x^5+3x^3-x+1=0$$
It has one real and four complex roots and makes a nice convex pentagon in the complex plane (thanks, Wolfram Alpha):

Using the formula for the area of a convex polygon:
$$A=\frac{1}{2} \left( \begin{array}| x_1 &amp; x_2 \\ y_1 &amp; y_2 \end{array} + \begin{array}| x_2 &amp; x_3 \\ y_2 &amp; y_3 \end{array} + \dots +  \begin{array}| x_n &amp; x_1 \\ y_n &amp; y_1 \end{array} \right)$$
I obtained for this case (using numerical values of the roots):
$$A=1.460144\dots$$

Another simple case - roots of unity. They just make regular polygons and the general formula for the area is well known.


However, I would like to know if it's possible to find out this area without computing the roots, using only the coefficients of the polynomial? (The coefficients are meant to be rational).

I know that polynomials with only real roots will all have $A=0$, and for the polynomials with several real roots some of them will be inside our maximum area polygon.
There is a useful theorem (see Rouche's theorem ), according to which:

For a monic polynomial $$z^n+a_{n-1} z^{n-1}+\dots+a_1 z+a_0$$
All its roots will be located inside the circle $|z|=1+\max |a_k|$.

But this theorem gives relatively large area, and can't be used to approximate the area of the polygon.
i was trying to attempt your question but i do not understand how you got that area there, i worked mine on maxima.... is there something i missed?
$$A=\frac{1}{2} \left( \begin{array}| x_1 &amp; x_2 \\ y_1 &amp; y_2 \end{array} + \begin{array}| x_2 &amp; x_3 \\ y_2 &amp; y_3 \end{array} + \dots +  \begin{array}| x_n &amp; x_1 \\ y_n &amp; y_1 \end{array} \right)$$
$$2x^5+3x^3-x+1=0$$
$\text{allroots(%)}$
$$[x = 0.415380864154341i+0.4410774845527911,x = 0.4410774845527911-0.415380864154341i,x = -0.7513475303185956,x = 1.344822399888639i-0.06540371939349326,x = -1.344822399888639i-0.06540371939349326]$$
$$[y_5:  0.415380864154341, x_5: 0.4410774845527911,x_4: 0.4410774845527911, y_4 : -0.415380864154341,x_3: -0.7513475303185956, y_3: 0, y_2: 1.344822399888639, x_2: -0.06540371939349326, y_1: -1.344822399888639 , x_1 : -0.06540371939349326]$$
$$ A = \frac{x_4y_5-x_1y_5-x_5y_4+x_3y_4-x_4y_3+x_2y_3-x_3y_2+x_1y_2+x_5y_1-x_2y_1}{2}$$
$$A = 0.4735192335485968\dots$$
The polygon with vertices of roots of a polynomial must be  symmetric about real axis because of the conjugate roots.  Area of the polygon equals to 2 times the region in upper half plane.The formula of area of polygon must be abandoned if polygon area is hoped to be computed with coefficients. This is because that formula requires the vertices of polygon which is impossible to be computed by the coefficients of the polynomial if the degree of the polynomial is equals or higher than 5 due to Abel-Ruffini theorem.It is possible for us to find out area generated by polynomial of degree 3 or 4(1 and 2 do not give a region) with the coefficients.

Here are some hints.