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 \(Q(x, y, z)\). (i) Determine the order of the monomials in \(f\) for the three monomial orders \(\prec_{\text {lex }}\), \(\prec_{\text {grlex }}\), and \(\prec_{\text {grevlex, with }} x>y \succ 2\) in all cases. (ii) For each of the three monomial orders from (i), determine mdeg \((f), \operatorname{lc}(f), \operatorname{lm}(f)\), and \(\operatorname{lt}(f)\).

Short answer

For lex and grlex, the leading term is \(2x^4y^2z\); for grevlex, it is \(4xy^4z^2\).

Step-by-step solution

List Monomials and Coefficients of f

Identify each monomial and its coefficient in the polynomial function \(f\).- \(2x^4y^2z\) with coefficient 2- \(-6x^4yz^2\) with coefficient -6- \(4xy^4z^2\) with coefficient 4- \(-3xy^2z^4\) with coefficient -3- \(x^2y^4z\) with coefficient 1- \(-5x^2yz^4\) with coefficient -5.

Determine Monomial Order for Lexicographic Order (lex)

In lexicographic order, compare monomials first by highest degree in \(x\), then \(y\), then \(z\).- Order: \(x^4y^2z, x^4yz^2, x^2y^4z, x^2yz^4, xy^4z^2, xy^2z^4\)Thus, \(f\) in lex order is: \(2x^4y^2z - 6x^4yz^2 + x^2y^4z - 5x^2yz^4 + 4xy^4z^2 - 3xy^2z^4\).

Determine Monomial Order for Graded Lexicographic Order (grlex)

In graded lexicographic order, compare monomials by total degree, and then use lex order for ties.- Degrees: \(x^4y^2z: 7, x^4yz^2: 7, xy^4z^2: 7, xy^2z^4: 7, x^2y^4z: 7, x^2yz^4: 7\)- Order: \(x^4y^2z, x^4yz^2, x^2y^4z, x^2yz^4, xy^4z^2, xy^2z^4\).Thus, \(f\) in grlex order is: \(2x^4y^2z - 6x^4yz^2 + x^2y^4z - 5x^2yz^4 + 4xy^4z^2 - 3xy^2z^4\).

Determine Monomial Order for Graded Reverse Lexicographic (grevlex)

For graded reverse lexicographic order, compare by total degree then reverse lexicographic:- Use same total degree as in grlex (all 7)- Order: \(xy^4z^2, x^2y^4z, x^4y^2z, x^4yz^2, xy^2z^4, x^2yz^4\)Thus, \(f\) in grevlex order is: \(4xy^4z^2 + x^2y^4z + 2x^4y^2z - 6x^4yz^2 - 3xy^2z^4 - 5x^2yz^4\).

Determine Leading Monomial Properties for Lex Order

- Leading Monomial (\(LM\)): \(x^4y^2z\)- Leading Coefficient (\(LC\)): 2- Leading Term (\(LT\)): \(2x^4y^2z\)- Maximal Degree (\(mdeg\)): 7

Determine Leading Monomial Properties for Grlex Order

- Leading Monomial (\(LM\)): \(x^4y^2z\)- Leading Coefficient (\(LC\)): 2- Leading Term (\(LT\)): \(2x^4y^2z\)- Maximal Degree (\(mdeg\)): 7

Determine Leading Monomial Properties for Grevlex Order

- Leading Monomial (\(LM\)): \(xy^4z^2\)- Leading Coefficient (\(LC\)): 4- Leading Term (\(LT\)): \(4xy^4z^2\)- Maximal Degree (\(mdeg\)): 7

Key concepts

Lexicographic Order

The lexicographic order, often abbreviated as 'lex' order, is a method of ranking monomials similar to the way words are ordered in a dictionary. In this system, you compare the powers of variables starting from a specific priority order. For instance, in this problem, the priority order is set as \(x > y > z\). Hence, you first compare the exponents of \(x\), then \(y\), and finally \(z\).
Let's take an example from the polynomial provided: To determine the order between \(x^4 y^2 z\) and \(x^4 y z^2\), observe that the power of \(x\) is the same in both monomials. Therefore, we move to the next variable in priority, which is \(y\). Here, the monomial \(x^4 y^2 z\) has a higher power of \(y\) compared to \(x^4 y z^2\), thus \(x^4 y^2 z\) is considered larger.
Using lexicographic order simplifies the comparison process, especially when the degrees of the initial variable differ. This method helps keep polynomials organized and aids in determining the leading monomial, coefficient, and term.

Graded Lexicographic Order

The graded lexicographic order, abbreviated as "grlex", offers a mix of total degree sorting with lexicographic comparisons. Here’s how it works: first, we sum the exponents of the monomials, and compare their total degree. If two or more monomials share the same total degree, we then use the lexicographic order to break the tie based on their variable priorities.
In our polynomial, all monomials share a maximum degree of 7. This requires us to fall back on lexicographic criteria as explained earlier. For example, the monomials \(x^4 y^2 z\) and \(x^2 y^4 z\) both amass a total degree of 7. In this case, the lexicographic order favors \(x^4 y^2 z\) as before, due to its higher power in \(x\). This dual-level sorting offers a more dynamic way of ranking monomials than regular lexicographic order.
Grlex order is highly useful in polynomial algebra, particularly in computational applications such as Gröbner bases, where efficient sorting of terms by degree helps streamline algebraic manipulations.

Graded Reverse Lexicographic Order

Graded reverse lexicographic order, or "grevlex", provides yet another way to sort monomials—it combines total degree ordering with a twist on lexicographic order. As always, begin by organizing monomials by their total degree. In case of a tie, evaluate using the reverse lexicographic method.
The reverse lexicographic approach differs by comparing variables from the least significant (or rightmost) to the most significant (or leftmost). For example, in the polynomial given, monomials like \(xy^4z^2\) and \(x^2y^4z\) possess equal total degrees. Using reverse lexicographic criteria, we compare variables starting from \(z\). Here, both have \(z\) raised to the power of 2 or more, pushing us to compare \(y\) and subsequently \(x\). In this order, \(xy^4z^2\) comes before \(x^2y^4z\).
Grevlex is particularly powerful in designing algorithms for computational algebra systems, as it efficiently navigates multiple terms sharing identical total degrees. Its reverse nature helps in detecting intrinsic relationships between monomials, making it a favorite in symbolic computation fields.