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Alternating polynomial
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<p>In algebra, an alternating polynomial is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} such that if one switches any two of the variables, the polynomial changes sign: </p> f ( x 1 , … , x j , … , x i , … , x n ) = − f ( x 1 , … , x i , … , x j , … , x n ) . {\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}).} <p>Equivalently, if one <a href="/facts/Permutes/JSw9ukmv">permutes</a> the variables, the polynomial changes in value by the <a href="/facts/Sign_of_a_permutation/ludhCMJz">sign of the permutation</a>: </p> f ( x σ ( 1 ) , … , x σ ( n ) ) = s g n ( σ ) f ( x 1 , … , x n ) . {\displaystyle f\left(x_{\sigma (1)},\dots ,x_{\sigma (n)}\right)=\mathrm {sgn} (\sigma )f(x_{1},\dots ,x_{n}).} <p>More generally, a polynomial f ( x 1 , … , x n , y 1 , … , y t ) {\displaystyle f(x_{1},\dots ,x_{n},y_{1},\dots ,y_{t})} is said to be <i>alternating in</i> x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} if it changes sign if one switches any two of the x i {\displaystyle x_{i}} , leaving the y j {\displaystyle y_{j}} fixed. </p>