Elementary symmetric polynomials are a special class of polynomials that arise in algebra, particularly in the study of polynomial roots. For a set of variables x_1, x_2, \ldots, x_n , the k -th elementary symmetric polynomial, denoted as e_k(x_1, x_2, \ldots, x_n) , is defined as 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 pairs of variables. These polynomials play a crucial role in various areas of mathematics, including combinatorics and representation theory. They are particularly important in symmetric functions and can be used to express the roots of polynomials in terms of their coefficients, as described by Vieta's formulas. Elementary symmetric polynomials provide a systematic way to