Alternating polynomial ($P(x_{\sigma (1)},\dots ,x_{\sigma (m)})=\mathrm{sgn}(\sigma )P(x)$)

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.

A monomial alternant is the determinant formed from powers with an integer exponent tuple $\ell$. It is a polynomial when the exponents are nonnegative, and otherwise a Laurent polynomial. Equal exponents give equal columns and zero determinant; interchanging exponents changes its sign. Ordered strictly decreasing nonnegative exponents give a basis of alternating polynomials, by grouping monomials into permutation orbits.