  This work is concerned with the study of the extreme rays of the convex cone
of $3\times 3$ quasiconvex quadratic forms (denoted by ${\cal C}_3$). We
characterize quadratic forms $f\in {\cal C}_3,$ the determinant of the acoustic
tensor of which is an extremal polynomial, and conjecture/discuss about other
cases. We prove that in the case when the determinant of the acoustic tensor of
a form $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square,
then the form must itself be an extreme ray of ${\cal C}_3;$ when the
determinant is a perfect square, then the form is either an extreme ray of
${\cal C}_3$ or polyconvex; and finally, when the determinant is identically
zero, then the form $f$ must be polyconvex. The zero determinant case plays an
important role in the proofs of the other two cases. We also make a conjecture
on the extreme rays of ${\cal C}_3,$ and discuss about weak and strong etremals
of ${\cal C}_d$ for $d\geq 3.$ where it turns out that several properties of
${\cal C}_3$ do not hold for ${\cal C}_d$ for $d>3,$ and thus case $d=3$ is
special. These results recover all previously known results (to our best
knowledge) on examples of extreme points of ${\cal C}_3$ that were proved to be
such. Our results also improve the ones proven by the first author and Milton
[20].
