Question

Which of the following finite subsets of \(\mathbb{Q}[x, y, z]\) are Gröbner bases with respect to \(\prec=\prec\) lex? Which are minimal or even reduced? (i) \(\left\\{x+y, y^{2}-1\right\\}\) for \(x \succ y\), (ii) \(\left\\{y+x, y^{2}-1\right\\}\) for \(y \succ x\), (iii) \(\left\\{x^{2}+y^{2}-1, x y-1, x+y^{3}-y\right\\}\) for \(x \succ y\), (iv) \(\left\\{x y z-1, x-y, y^{2} z-1\right\\}\) for \(x \succ y \succ z\).

Short answer

Sets (i) and (ii) are reduced Gröbner bases; (iii) and (iv) are not.

Step-by-step solution

Analyze polynomial set (i)

The set \( \{ x+y, y^2-1 \} \) is for order \( x \succ y \). Check leading terms. The leading term of \( x+y \) is \( x \), and for \( y^2-1 \) it's \( y^2 \). Since they involve different variables sequentially ordered in \( x \succ y \), they form a Gröbner basis. Verify minimality: both polynomials are needed for the leading terms. To check if it's reduced, the polynomials have no monomial that's redundant beyond leading terms.

Analyze polynomial set (ii)

The set \( \{ y+x, y^2-1 \} \) is for order \( y \succ x \). The leading term of \( y+x \) is \( y \), and for \( y^2-1 \) it's \( y^2 \). After dividing terms, \( y \) complements \( y^2 \) as needed, maintaining a Gröbner basis. To check the minimality, similar reasons confirm its minimal and checking reduced idx shows no derivations indicate redundancy with term coefficients.

Analyze polynomial set (iii)

The set \( \{ x^2+y^2-1, xy-1, x+y^3-y \} \) sets \( x \succ y \). The leading term of each polynomial \( x^2, xy, \) and \( x \), covers terms sufficiently diverse to form a Gröbner basis. Check minimality by examining unnecessary polynomials. \( xy-1 \) and \( x \), check reduced by analyzing no cancels of similar monomials unless lead terms.

Analyze polynomial set (iv)

The set \( \{ xyz-1, x-y, y^2z-1 \} \) for order \( x \succ y \succ z \). The leading term of each \( xyz, x, y^2z \) maintains diversity for a potential Gröbner basis. Testing implies S-polynomial reduction validation may hold leading terms diverse and disjoint. Verify terms have no common factor leading to unnecessary elements, confirming reduced form is already simplified.

Key concepts

Understanding Polynomials

Polynomials are mathematical expressions involving variables and coefficients, joined together using operations like addition and multiplication.
They can have one or more terms, such as \( x^2 + y + 1 \) being a polynomial with three terms. In the context of Gröbner bases, polynomials serve as building blocks for forming ideals in a polynomial ring.

When dealing with polynomials in this context, we often use complex algebraic structures like multivariate polynomials, which involve more than one variable. This allows us to explore behaviors and interactions between different variables.

• Each term in a polynomial is made up of a product of coefficients and variables raised to some power.
• The degree of a polynomial is determined by the term with the highest total exponent.
• Understanding how polynomials are structured helps in categorizing and solving polynomial systems.

They are fundamental in both pure and applied mathematics, appearing in everything from solving equations to modeling real-world phenomena.

Leading Terms in Polynomials

The concept of the *leading term* in a polynomial is crucial when dealing with Gröbner bases. The leading term of a polynomial is defined as the term with the highest degree based on a specified ordering in its variables.
This term exhibits the polynomial's "dominant" behavior in computations and simplifications, particularly when considering polynomial division.
The leading term often guides decision-making in algebraic processes, for instance:

• Identifying key factors in polynomial identities.
• Determining ordering in a reduction or simplification process.
• Playing a pivotal role in the reduction of polynomial systems into Gröbner bases.

For a polynomial like \( x^2 + 3xy + y^2 \), if variables are ordered with \( x \succ y \), the leading term is \( x^2 \). This dictates the main term against which others are compared during various operations, mainly when checking membership in an algebriac ideal.

Lexicographic Order in Polynomials

Lexicographic order is a way to arrange the terms of polynomials, similar to how words are ordered in a dictionary.
This ordering is significant in forming Gröbner bases, where choosing the correct term order can drastically simplify calculations and derivations.

In lexicographic ordering, variables are prioritized according to a preset hierarchy. For example, if we decide \( x \succ y \succ z \), any polynomial term involving \( x \) will always precede those only involving \( y \) or \( z \). This results in:

• Order decisions that inform polynomial reduction and simplification strategies.
• Ensuring consistency across computations within Gröbner bases.
• Guiding the algebraic algorithm in checking polynomial independence.

Using lexicographic ordering aligns the polynomial terms against an objective rule, aiding in establishing fair comparisons in complex algebraic systems. Understanding this ordering ensures that students use the same assumptions in polynomial manipulation.

Redundancy in Polynomials

Redundancy in polynomials refers to unnecessary or repetitive terms or factors within a set of polynomials.
The idea is to simplify polynomial expressions or bases by removing redundant elements that do not contribute unique information.
This becomes particularly important in the context of Gröbner bases, as maintaining a minimal set of polynomials ensures efficiency in computations.

When a polynomial set is free of redundancy, it implies that no polynomial can be removed without losing the generation ability of the basis.

• Assess each polynomial's necessity in terms of contributing to the leading coefficients and terms of the basis.
• Ensure that no polynomial is a combination of others in the set.
• Verify that reduction processes uphold a basis's properties without unnecessary complexity.

By minimizing redundancy, students and mathematicians alike ensure that they work with the simplest and most efficient model of a polynomial system, maximizing performance in solving algebraic problems.