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

## Synopsis

• Usage:
vandermondeDeterminant(l,R)
• Inputs:
• R, ,
• l, a list, a subset of the indices of the generators of R
• Optional inputs:
• AsExpression => , 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:
• , 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 .