A symmetric polynomial on n variables x_1, ..., x_n (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy f(y_1, y_2, ..., y_n) = f(x_1, x_2, ..., x_n), where y_i = x_(π(i)) and π being an arbitrary permutation of the indices 1, 2, ..., n. For fixed n, the set of all symmetric polynomials in n variables forms an algebra of dimension n. The coefficients of a univariate polynomial f(x) of degree n are algebraically independent symmetric polynomials in the roots of f, and thus form a basis for the set of all such symmetric polynomials.
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