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Vandermonde Determinant

$\displaystyle \Delta(x_1, \ldots, x_n)$ $\textstyle \equiv$ $\displaystyle \left\vert\begin{array}{ccccccccccccccccc}
1 & x_1 & {x_1}^2 & \c...
1 & x_n & {x_n}^2 & \cdots & {x_n}^{n-1}\end{array}\right\vert\nonumber$  
  $\textstyle =$ $\displaystyle \prod_{\scriptstyle i,j\atop \scriptstyle i>j} (x_i-x_j)$  

(Sharpe 1987). For Integers $a_1$, ..., $a_n$, $\Delta(a_1, \ldots, a_n)$ is divisible by $\prod_{i=1}^n (i-1)!$ (Chapman 1996).

See also Vandermonde Matrix


Chapman, R. ``A Polynomial Taking Integer Values.'' Math. Mag. 69, 121, 1996.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1111, 1979.

Sharpe, D. §2.9 in Rings and Factorization. Cambridge, England: Cambridge University Press, 1987.

© 1996-9 Eric W. Weisstein