Algebra/Chapter 10/Symmetric Polynomials/Monomial Ordering

Definition 1 Let be an N-tuple. Let us define: For functions let us define: This abbreviated notation will be of great use to us in the following pages and in the proof. Definition 2 Let be a field. Then is the polynomial space in variables with coefficients in . A monomial is a polynomial of the form , such that and . Definition 3 Let , and let be an exponent vector. Let us define: The degree of a non-zero polynomial is equal to the maximum of the degrees of its compsing monomials. Properties Monomial multiplication maintains exponent vector addition: Definition 4 Let be monomials. We say that is of lower order than (and denote it by ) if there exists an index such that In other words, the vectors have a lexicographic ordering. In a polynomial , the monomial of maximal order is called the leading monomial, and is denoted by . Example Lemma Let be polynomials. Then . Proof Let be monomials, with . 1. Let us assume that . We will show that for all . By definition, there exists an index such that 2. Let us assume also that . We will show that . By definition, there exist indexes such that respectively hence: Definition 5 Let be a polynomial. Let us define: Meaning, the set of all monic monomials of degree which are of lower order than . Example Monomial Ordering Introduction Permutations and Symmetry