Recall the lexicographic monomial order with x1 x2 X defined

Last updated: 2/28/2023

Recall the lexicographic monomial order with x1 x2 X defined in Section 9 6 where the nonzero monomial term with exponents a1 a2 an comes before the nonzero monomial term with exponents b1 b2 bn if the initial components of the two n tuples of exponents are equal and the first component where they differ has a bi If f x xn contains the monomial Axx2 x then since f x xn is symmetric it also contains all the permuted monomials Among these choose the lexicographically largest monomial which therefore satisfies a 2a22 an 0 38 a Show that the monomial As 222 23 s in the elementary symmetric functions has the same lexicographic initial term b Show that subtracting As 22 3 2 s from f x yields either 0 or a symmetric polynomial of the same degree whose terms are lexicographically smaller than the terms in f x1 Xn c Show that the iteration of this procedure lexicographic ordering choosing the lex icographically largest term subtracting the associated monomial in the elementary symmetric functions terminates expressing f x1 xn as a polynomial in the elementary symmetric functions