14.1 Monomial orderings and division algorithm

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 well-ordering 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
  • $x_1^{a_1} \cdots x_n^{a_n} < x_1^{b_1} \cdots x_n^{b_n}$ if $a_i < b_i$ for leftmost $i$ where $a_i \neq b_i$
  • Example: $x^2y < xy^3$ 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
  • $x_1^{a_1} \cdots x_n^{a_n} < x_1^{b_1} \cdots x_n^{b_n}$ if:
    1. $\sum a_i < \sum b_i$, or
    2. $\sum a_i = \sum b_i$ and $x_1^{a_1} \cdots x_n^{a_n} <_{lex} x_1^{b_1} \cdots x_n^{b_n}$
  • Example: $xy^2 < x^2y$ 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
  • $x_1^{a_1} \cdots x_n^{a_n} < x_1^{b_1} \cdots x_n^{b_n}$ if:
    1. $\sum a_i < \sum b_i$, or
    2. $\sum a_i = \sum b_i$ and $a_j > b_j$ for rightmost $j$ where $a_j \neq b_j$
  • Example: $x^2y < xy^2$ in grevlex because total degrees are equal, and $y$ has lower priority

Division Algorithm

Describe the division algorithm for multivariate polynomials

• Division algorithm generalizes univariate polynomial division to multivariate case
• Purpose divides polynomial $f$ by set of polynomials $F = \{f_1, \ldots, f_s\}$ using specified monomial ordering
• Input requires polynomial $f$, set of polynomials $F$, and chosen monomial ordering
• Output produces quotients $q_1, \ldots, q_s$ and remainder $r$ satisfying $f = q_1f_1 + \cdots + q_sf_s + r$
• Algorithm steps:
  1. Initialize quotients to zero and remainder to $f$
  2. While remainder is not zero:
     • Find first $f_i$ whose leading term divides leading term of remainder
     • If found, subtract appropriate multiple from remainder and add to quotient
     • If not found, move leading term of remainder to $r$
  3. Return quotients and remainder
• Example: Dividing $f = x^2y + xy^2 + y^2$ by $F = \{f_1 = xy - 1, f_2 = y^2 - 1\}$ 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:
  • Gröbner basis 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 reduction:
  • 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