Alternating polynomials

In algebra, an alternating polynomial is a polynomial ${\displaystyle f(x_{1},\dots ,x_{n})}$ such that if one switches any two of the variables, the polynomial changes sign:

Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation:

More generally, a polynomial ${\displaystyle f(x_{1},\dots ,x_{n},y_{1},\dots ,y_{t})}$ is said to be alternating in ${\displaystyle x_{1},\dots ,x_{n}}$ if it changes sign if one switches any two of the ${\displaystyle x_{i}}$, leaving the ${\displaystyle y_{j}}$ fixed.

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