Take the 2-minute tour ×  Three points $z_1$, $z_2$, $z_3$ on
the complex plane are given by the coefficients $a_k$'s of the
cubic polynomial $f(z)=(z-z_1)(z-z_2)(z-z_3)=\sum_{k=0}^3 a_k
z^k$. How does one express the (signed) area $V$ of the triangle
with vertices $z_1$, $z_2$, $z_3$ in terms of $a_k$'s and
$\overline{a}_k$'s? One is tempted to try to expand $V^2$ in the
symmetric functions in the roots of $f(z)\overline{f}(z)$, as
well as these of $f(z)$ and of $\overline{f}(z)$, e.g. starting
from $$ V=\frac{\sqrt{-1}}{4}\det \begin{pmatrix} 1&amp; 1&amp;1
\&bsol; z_1&amp;z_2&amp;z_3 \&bsol; \overline{z}_1&amp;\overline
{z}_2&amp;\overline{z}_3\end{pmatrix}, $$ (this not so well-
known formula can be found in R.Deaux, Introduction to the
Geometry of Complex Numbers, Ungar, New York, 1956, pp.59-60),
but this rather calls for some kind of joint invariants of $f$
and $\overline{f}$ to be used.  Any pointers etc. are much
appreciated.  Added: the motivation comes from a moment problem:
suppose one is given a pa