This is all true, but what interpretation/meaning should we give to the square of the Vandermonde determinant? For example, what meaning would this number have in the case that the three roots of a cubic are the vertices of a triangle? If the area of the triangle is zero, the discriminant is still non-zero, so if the discriminant is a function of the area, then it must somewhat complicated. I was wondering if the discriminant could be some function of other properties of the triangle (of the roots). At 12:43 PM 7/1/2015, rcs@xmission.com wrote:

Geometric Interpretation of the Discriminant

Following up Gene's note on Discriminant = square of Vandermonde determinant ... The determinant is +- the area/volume/measure of a parallelogram/piped/? with origin 0 and each row (or each column) as a generating edge. The determinant of the matrix

[ 1 2 ] [ 3 4 ]

is -2, which is also the area of the parallelogram 00 - 12 - 46 - 34. (The +- depends on your sign convention, and is extinguished if you square the Vandermonde determinant.) The determinant of a V matrix is 0 whenever two roots of the related polynomial are equal.