Elementary symmetric functions are mathematical expressions that represent sums of products of variables. For a set of variables x_1, x_2, \ldots, x_n , the k -th elementary symmetric function, denoted as e_k , is the sum of all possible products of k distinct variables. For example, e_1 is the sum of the variables, while e_2 is the sum of the products of all pairs of variables. These functions play a crucial role in various areas of mathematics, including algebra, combinatorics, and polynomial theory. They are particularly important in the study of polynomial roots and symmetric polynomials, as they provide a way to express the relationships between the roots of a polynomial and its coefficients.