vandermondeDeterminant -- the vandermonde determinant for a set of generators of a ring

Synopsis

Usage:

vandermondeDeterminant(l,R)
Inputs:
- R, a polynomial ring,
- l, a list, a subset of the indices of the generators of R
Optional inputs:
- AsExpression => a Boolean value, default value false, a Boolean value, default value is false. If true it returns the determinant as a product expression This is a particularly useful way to reduce the size of the object since a Vandermonde determinant has n! terms but only n*(n-1)/2 factors.
Outputs:
- a ring element, the determinant of the Vandermonde matrix formed by the generators indexed by l.

Description

A Vandermonde matrix is a matrix of $n$ elements is constructed by putting in each column all the powers from 0 to $n-1$ of each of the elements.

If $x_i$ are the elements used to construct the matrix then it can be proven that the determinant has the following form.

$\prod_{0 \leq i < j < n} (x_j-x_i) $

i1 : R = QQ[x_0..x_3]

o1 = R

o1 : PolynomialRing

i2 : vandermondeDeterminant({0,2,3},R)

        2        2    2      2        2      2
o2 = - x x  + x x  + x x  - x x  - x x  + x x
        0 2    0 2    0 3    2 3    0 3    2 3

o2 : R

i3 : factor oo

o3 = (x  - x )(x  - x )(x  - x )(-1)
       2    3   0    3   0    2

o3 : Expression of class Product

Ways to use vandermondeDeterminant :

vandermondeDeterminant(List,PolynomialRing)

For the programmer

The object vandermondeDeterminant is a method function with options.