Question

Compute a Grobbner basis for $$ \left(f_1=x^2 y-2 y z+1, f_2=x y^2-z^2+2 x, f_3=y^2 z-x^2+5\right) \subseteq \mathbb{Q}[x, y, z], $$ using \(\prec=\prec_{\text {grles }}\) with \(x \prec y \prec z\). Compare your output to the Gr?bner basis that MAPLE computes with a different order.

Short answer

Use Buchberger's Algorithm to compute the Groebner basis, comparing the output with MAPLE’s result under a different order.

Step-by-step solution

Define the Polynomial Ring

Consider the polynomial ring \(\mathbb{Q}[x, y, z]\) where \(x \prec y \prec z\) indicates a graded lexicographical order with \(x\) as the smallest element. We are given polynomials \(f_1 = x^2 y - 2yz + 1\), \(f_2 = xy^2 - z^2 + 2x\), and \(f_3 = y^2 z - x^2 + 5\).

Initiate the Buchberger's Algorithm

We use Buchberger's algorithm to compute a Groebner basis. Start with the set \(\{f_1, f_2, f_3\}\) and compute the S-polynomials for each pair: \(S(f_i, f_j) = \text{reduce}\left(\frac{\text{LCM}\,(\text{LT}(f_i), \text{LT}(f_j))}{\text{LT}(f_i)}f_i - \frac{\text{LCM}\,(\text{LT}(f_i), \text{LT}(f_j))}{\text{LT}(f_j)}f_j\right)\), where \(\text{LT}\) denotes the leading term.

Compute S-Polynomials

Compute the S-polynomials for each pair of polynomials:* \(S(f_1, f_2)\)* \(S(f_1, f_3)\)* \(S(f_2, f_3)\)Simplify each S-polynomial to see if it reduces to zero. Any non-zero result should be added to the basis.

Add New Basis Elements

If any S-polynomial does not reduce to zero, add the resulting polynomial to your current basis set. Repeat the process until no new polynomials are added, yielding a Groebner basis.

Verify the Groebner Basis

Once no more S-polynomials result in non-zero polynomials, the remaining set is your Groebner basis for the given polynomials under the lexicographical order \(x \prec y \prec z\).

Comparing the Results with MAPLE

Run the same computation in MAPLE using its Groebner basis computation tool but under a different monomial order. Compare the two resulting Groebner bases to analyze how the choice of monomial order in some cases affects the output.

Key concepts

Buchberger's Algorithm

When dealing with polynomial ideals, Buchberger's Algorithm is an essential tool. It allows for the computation of a Groebner basis, which is a set of polynomials that has similar properties to simpler algebraic structures. This algorithm involves iteratively calculating what's known as S-polynomials from a given set of generating polynomials.
To apply Buchberger’s Algorithm, you start with an initial set of polynomials. In our example, these are \(f_1, f_2,\) and \(f_3\). Using these, you compute the S-polynomials and simplify them through division by the leading terms of the polynomials involved. This process continues until no new polynomials are added, ensuring the final set represents the Groebner basis.
Buchberger’s Algorithm is powerful because it gives a systematic way to address polynomial ideal problems, yet it can get computationally intensive for complex systems or higher dimensions.

Polynomial Ring

A polynomial ring is a mathematical construct where polynomials form a ring under addition and multiplication operations. Rings are fundamental structures in algebra used to generalize concepts from arithmetic.
For this problem, the polynomial ring is \(\mathbb{Q}[x, y, z]\), which means it consists of polynomials with variables \(x, y, z\) and coefficients in the field of rational numbers, \(\mathbb{Q}\). This setup is critical as the ring defines how the operations on the polynomials are governed and impacts the outcome of processes like finding a Groebner basis.
Understanding the polynomial ring involved is essential because it dictates key properties and behaviors of polynomial operations, such as zero divisors and invertibility, hence it's foundational when working with concepts like Buchberger's Algorithm.

Graded Lexicographical Order

In Groebner basis computations, the ordering of terms is crucial to determine leading terms and control the division process. A graded lexicographical order is a specific type of term order. It considers both the total degree of polynomials and a lexicographical comparison for terms of the same degree.
For the provided polynomial set, the order \(x \prec y \prec z\) means when comparing two polynomials, you prioritize their degree total weight first, then use the standard dictionary-like order for terms of the same degree. This ensures that the computation of Groebner basis proceeds deterministically.
Choosing such an order affects the shape and number of polynomials in the Groebner basis. Therefore, different orders may lead to distinct bases and computations under tools like MAPLE. This priority system also often simplifies interpretation since leading terms directly project priority.

S-polynomials

S-polynomials are central to Buchberger's Algorithm. They are derived to eliminate leading terms in pairs of polynomials, guiding the construction of a Groebner basis by checking for and resolving non-monotone behavior in polynomial reductions.
To compute an S-polynomial for two polynomials \(f_i\) and \(f_j\), you first find the least common multiple (LCM) of their leading terms, \(\text{LT}(f_i)\) and \(\text{LT}(f_j)\). This LCM helps align both polynomials to cancel out their leading terms when subtracted. The formula is: \[S(f_i, f_j) = \frac{\text{LCM}(\text{LT}(f_i), \text{LT}(f_j))}{\text{LT}(f_i)}f_i - \frac{\text{LCM}(\text{LT}(f_i), \text{LT}(f_j))}{\text{LT}(f_j)}f_j\]
Any resulting polynomial that is non-zero upon simplification should be included in the set, refining the basis each step. This iterative process continues until all S-polynomials lead to the construction of the final Groebner basis.