= Alternating polynomial
{title2=$P(x_{\sigma(1)},\ldots,x_{\sigma(m)})=\operatorname{sgn}(\sigma)P(x)$}
Over a field of <characteristic zero>, an alternating <polynomial> changes sign when two variables are interchanged. It vanishes when two coordinates agree, so each factor $x_i-x_j$ divides it. These distinct linear factors are relatively prime in the <polynomial ring>, hence their product, the <Vandermonde determinant>, divides it. The quotient is a <symmetric polynomial>.
Back to article page
