Alternating polynomial

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:

${\displaystyle f(x_{1},\dots ,x_{j},\dots ,x_{i},\dots ,x_{n})=-f(x_{1},\dots ,x_{i},\dots ,x_{j},\dots ,x_{n}).}$

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

${\displaystyle f\left(x_{\sigma (1)},\dots ,x_{\sigma (n)}\right)=\mathrm {sgn} (\sigma )f(x_{1},\dots ,x_{n}).}$

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.^[1]