Question

Let \(f=2 x^{4} y^{2} z-6 x^{4} y z^{2}+4 x y^{4} z^{2}-3 x y^{2} z^{4}+x^{2} y^{4} z-5 x^{2} y z^{4}\) in \(\mathbb{Q}[x, y, z]\). (i) Determine the order of the monomials in \(f\) for the three monomial orders \(\prec_{\text {lex }}, \prec_{\text {grlex }}\), and \(\prec_{\text {grevlex }}\), with \(x \succ y \succ z\) in all cases. (ii) For each of the three monomial orders from (i), determine \(\operatorname{mdeg}(f), \operatorname{lc}(f), \operatorname{lm}(f)\), and \(\operatorname{lt}(f)\).

Short answer

The order of monomials differs by the monomial order: lex favors \(2x^4y^2z\), grlex and grevlex favor \(4xy^4z^2\). The leading terms match these orders.

Step-by-step solution

Define Monomial Orders

In the lexicographic order (lex), monomials are ordered first by the degree of x, then y, and finally z. In the graded lexicographic order (grlex), monomials are compared first by total degree, and if tied, by lex order. In the graded reverse lexicographic order (grevlex), monomials are compared first by total degree, and if tied, by lex order, but the last variable has priority in the tie-breaking process.

Identify Monomials in f

The given polynomial f has the following monomials: \(2x^4y^2z\), \(-6x^4yz^2\), \(4xy^4z^2\), \(-3xy^2z^4\), \(x^2y^4z\), and \(-5x^2yz^4\).

Arrange Monomials in Lexicographic Order

Order the monomials in f using \(\prec_{\text{lex}}\): \(2x^4y^2z \succ -6x^4yz^2 \succ x^2y^4z \succ 4xy^4z^2 \succ -3xy^2z^4 \succ -5x^2yz^4\).

Arrange Monomials in Graded Lexicographic Order

The total degree of each monomial enables their ordering by \(\prec_{\text{grlex}}\):\(4xy^4z^2\) (degree 7) \succ \(-3xy^2z^4\) (degree 7) \succ \(2x^4y^2z\) (degree 7) \succ \(-6x^4yz^2\) (degree 7) \succ \(x^2y^4z\) (degree 7) \succ \(-5x^2yz^4\) (degree 7). The tie is broken by lex order.

Arrange Monomials in Graded Reverse Lexicographic Order

The total degree and ordering by \(\prec_{\text{grevlex}}\): \(4xy^4z^2\) (degree 7) \succ \(-3xy^2z^4\) (degree 7) \succ \(2x^4y^2z\) (degree 7) \succ \(-6x^4yz^2\) (degree 7) \succ \(x^2y^4z\) (degree 7) \succ \(-5x^2yz^4\) (degree 7). The tie is broken with z's power descending.

Determine mdeg(f), lc(f), lm(f), and lt(f) for Lex Order

Using \(\prec_{\text{lex}}\), the leading monomial is \(2x^4y^2z\) with these characteristics: \(\operatorname{mdeg}(f) = (4,2,1)\), \(\operatorname{lc}(f) = 2\), \(\operatorname{lm}(f) = x^4y^2z\), and \(\operatorname{lt}(f) = 2x^4y^2z\).

Determine mdeg(f), lc(f), lm(f), and lt(f) for Grlex Order

Using \(\prec_{\text{grlex}}\), the first monomial is \(4xy^4z^2\) with: \(\operatorname{mdeg}(f) = (1,4,2)\), \(\operatorname{lc}(f) = 4\), \(\operatorname{lm}(f) = xy^4z^2\), and \(\operatorname{lt}(f) = 4xy^4z^2\).

Determine mdeg(f), lc(f), lm(f), and lt(f) for Grevlex Order

Using \(\prec_{\text{grevlex}}\), \(4xy^4z^2\) is also the leading monomial, yielding: \(\operatorname{mdeg}(f) = (1,4,2)\), \(\operatorname{lc}(f) = 4\), \(\operatorname{lm}(f) = xy^4z^2\), and \(\operatorname{lt}(f) = 4xy^4z^2\).

Key concepts

Graded Lexicographic Order

In algebra, the Graded Lexicographic Order (often abbreviated as Grlex) is a method for sorting monomials in a polynomial. When using this order, you first consider the total degree of each monomial. The total degree is the sum of the exponents in a monomial. Monomials with a higher total degree are given priority.

If two monomials have the same total degree, the tie is resolved using the Lexicographic Order. Here, the order of variables is significant. The monomial with the higher exponent of the first variable deemed most important in the variable hierarchy, is placed first.

For example, if we have variables such that \(x \succ y \succ z\), a tie between \(x^3yz, x^2y^2z^2\) (both having a total degree of 6), would be resolved by Grlex in favor of \(x^3yz\), because the x exponent is higher. This makes the Graded Lexicographic Order an efficient way to organize monomials, considering both the significance of the variable arrangement and the degree sum.

Lexicographic Order

Lexicographic Order is one of the simplest ways to sequence monomials, akin to how words are sorted in a dictionary. Each variable is given a rank, and in decreasing order of rank, each monomial is compared based on variable exponents. This means that priority is given to exponents of the 'highest ranked' variables first.

To better understand, let’s consider a scenario where the variable ranking is \(x \succ y \succ z\). When comparing two monomials, say \(x^5y^2\) and \(x^3y^4z\), the order is determined by their x powers first. Since \(x^5y^2\) has a greater x power compared to \(x^3y^4z\), it takes precedence regardless of the degrees of \(y\) or \(z\).

Lexicographic Order is straightforward and requires no computation of total degrees, allowing quick identification of dominant monomials.

Graded Reverse Lexicographic Order

The Graded Reverse Lexicographic Order (Grevlex) is a fascinating twist on the Graded Lexicographic Order. It builds on the idea of comparing monomials first by their total degree, similar to Grlex. However, it diverges when resolving ties between monomials having the same degree.

In Grevlex, ties are resolved by focusing on the rightmost or the last variable within the hierarchy. This means that if two monomials have identical total degrees, they are ordered based on the lowest-ranking variable, looking for greater exponents in this trailing variable.

For example, given variables \(x \succ y \succ z\), consider the monomials \(x^2yz^3\) and \(x^2y^3z\). While they share the same total degree, Grevlex would order \(x^2yz^3\) before \(x^2y^3z\) since the z power (3) is highest in the first monomial. This strategy of ordering grants a unique way to reconsider the importance of variables, allowing more nuanced arrangements of polynomial terms.