Graded lex and graded reverse lex I just want to know what the difference between graded lex and graded reverse lex is. For example, if we have
\begin{equation}
f=3x^4 y - 2x^2 y^3 z + 7x^2 y z^3 - x^3 y^2 z + z^3 y^3 - 6x^3 yz^2
\end{equation}
I know that if we have a graded rev lex with z>x>y we have that the lead term (LT) of $f$ will be $7x^2 y z^3$, what would LT$(f)$ be with graded lex?
 A: Both are graded, so for both you choose the largest total degree first, the lex and revlex stuff is only used if you need to break a tie.
If $y < x < z$ then for lex you look at the variables in the order $z \to x \to y$ (largest to smallest) and choose the largest degree, moving on to the next variable if you need to break a tie.  So among those terms with total degree $6$ (the largest) we choose $7x^2yz^3$ because we look at $z$ first and choose the term with the largest $z$-degree.
If we still have $y < x < z$ then for revlex you look at the variables in the order $y \to x \to z$ (smallest to largest) and you choose the smallest degree, moving on to the next variable if you need to break a tie.  So among those terms with total degree $6$ we would take $\mathrm{LT}(f) = 7x^2yz^3$ because the smallest $y$-degree you can get is $1$, and of the two monomials that have that $y$-degree the smallest $x$-degree you can get is $2$.
It's a little funny because the leading term comes out to be the same.  To see one that comes out different let $f = x^2 + zy$.  Keep $y < x < z$.  Then with graded lex we have $\mathrm{LT}(f) = yz$ and with graded revlex we have $\mathrm{LT}(f) = x^2$.
To keep them straight think that the "l" in lex is for large, and reverse large is small, so:


*

*lex $\to$ largest variable first, take largest degree

*revlex $\to$ smallest variable first, take smallest degree

A: That's right: one must inspect the monomials of highest total degree with indeterminats ordered by decreasing weights. For two monomials $m=z^{\alpha}x^{\beta}y^{\gamma}$ and $m'z^{\alpha'}x^{\beta'}y^{\gamma'}$, one has $m>m'$ if for the first differing exponent, starting from the right, $m$ has the smallest. So , by decreasing order, we have:
$$z^3x^2y\succ z^2x^3y\succ zx^3y^2\succ z^3y^3\succ zx^2y^3\succ x^4y$$
