Monomial orderings are crucial for comparing and arranging terms in polynomial rings. They define a total order on monomials, ensuring consistent comparisons. Key properties include and multiplicativity, which maintain order when multiplying monomials.
Common types of monomial orderings include lexicographic (lex), graded lexicographic (grlex), and graded reverse lexicographic (grevlex). Each has unique characteristics that affect how polynomials are ordered and manipulated in computational algebra algorithms.
Monomial Orderings
Define and explain monomial orderings
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Monomial ordering defines a total order on monomials in polynomial ring allowing comparison and arrangement
Properties of monomial orderings ensure consistent and well-defined ordering:
Well-ordering guarantees existence of smallest element in any nonempty set of monomials
Multiplicative property preserves order when multiplying monomials (if u<v, then uw<vw for any monomial w)
Common types of monomial orderings used in computational algebra:
Lexicographic (lex) order compares variables from left to right
Graded lexicographic (grlex) order considers total degree first, then lex order
Graded reverse lexicographic (grevlex) order uses total degree, then reverse lex order
Compare and contrast different types of monomial orderings
Lexicographic (lex) order prioritizes variables from left to right:
Compares exponents of variables sequentially
x1a1⋯xnan<x1b1⋯xnbn if ai<bi for leftmost i where ai=bi
Example: x2y<xy3 because x has higher priority
Graded lexicographic (grlex) order balances total degree and lex order:
First compares total degree of monomials
Uses lex order for monomials with equal total degree
x1a1⋯xnan<x1b1⋯xnbn if:
∑ai<∑bi, or
∑ai=∑bi and x1a1⋯xnan<lexx1b1⋯xnbn
Example: xy2<x2y in grlex because total degrees are equal, and x has higher priority
Graded reverse lexicographic (grevlex) order combines total degree and reverse lex:
Compares total degree first
Uses reverse lex order for equal total degrees
x1a1⋯xnan<x1b1⋯xnbn if:
∑ai<∑bi, or
∑ai=∑bi and aj>bj for rightmost j where aj=bj
Example: x2y<xy2 in grevlex because total degrees are equal, and y has lower priority
Division Algorithm
Describe the division algorithm for multivariate polynomials
generalizes univariate polynomial division to multivariate case
Purpose divides polynomial f by set of polynomials F={f1,…,fs} using specified monomial ordering
Input requires polynomial f, set of polynomials F, and chosen monomial ordering
Output produces quotients q1,…,qs and r satisfying f=q1f1+⋯+qsfs+r
Algorithm steps:
Initialize quotients to zero and remainder to f
While remainder is not zero:
Find first fi whose divides leading term of remainder
If found, subtract appropriate multiple from remainder and add to
If not found, move leading term of remainder to r
Return quotients and remainder
Example: Dividing f=x2y+xy2+y2 by F={f1=xy−1,f2=y2−1} using lex order with x>y
Explain the significance of the division algorithm in polynomial rings
Division algorithm extends univariate polynomial division to multivariate case
Provides method for reducing polynomials with respect to set of polynomials
Forms foundation for advanced computational algebra algorithms:
computation determines special generating set for polynomial ideals
Ideal membership testing checks if polynomial belongs to given ideal
Solving systems of polynomial equations finds common roots of multiple polynomials
Enables study of polynomial ideals and their properties (primary decomposition)
Demonstrates crucial role of monomial orderings in polynomial computations and algebraic geometry
Analyze the properties of the remainder in the division algorithm
Remainder properties ensure well-defined :
No term in r divisible by any leading term of polynomials in F
Degree of r less than or equal to degree of f
Non-uniqueness of remainder highlights dependency on:
Chosen monomial ordering affects term comparisons
Order of polynomials in F influences reduction process
Relation to ideal membership provides algebraic insights:
Zero remainder (r=0) implies f belongs to ideal generated by F
Non-zero remainder does not guarantee f is outside the ideal
Applications include simplification of polynomial expressions and computation of normal forms for algebraic varieties