Elementary symmetric polynomials are a special class of polynomials that arise in algebra, particularly in the study of symmetric functions. They are defined for a set of variables and represent sums of products of these variables taken in various combinations. For example, for three variables x_1, x_2, x_3, the elementary symmetric polynomials include e_1 = x_1 + x_2 + x_3, e_2 = x_1x_2 + x_1x_3 + x_2x_3, and e_3 = x_1x_2x_3. These polynomials play a crucial role in algebraic combinatorics and representation theory. They can be used to express any symmetric polynomial as a combination of elementary symmetric polynomials, which is a key concept in Newton's identities. The elementary symmetric polynomials are also closely related to the roots of {pol