UTM Ideals Varieties and Algorithm - Chapter 2 Section 2 Exercise 3

Problem:

Solution:

Here is the Exercise 1 statement.

Okay, I know this is stupid, this is just an exercise!

Part (a)
lex order: $ -z^2 + z + 3y + x^3 + x^2 + 2x $
$ LM(f) = z^2 $
$ LT(f) = -z^2 $
$ multideg(f) = (2, 0, 0) $
(2, 0, 0)
(1, 0, 0)
(0, 1, 0)
(0, 0, 3)
(0, 0, 2)
(0, 0, 1)

grlex order: $ x^3 - z^2  + x^2 + z + 3y + 2x $
$ LM(f) = x^3 $
$ LT(f) = x^3 $
$ multideg(f) = (0, 0, 3) $
(0, 0, 3)
(2, 0, 0)
(0, 0, 2)
(1, 0, 0)
(0, 1, 0)
(0, 0, 1)

grevlex order: $ x^3 + z^2 - x^2 + z + 3y + 2x $
$ LM(f) = x^3 $
$ LT(f) = x^3 $
$ multideg(f) = (0, 0, 3) $
(0, 0, 3)
(2, 0, 0)
(0, 0, 2)
(1, 0, 0)
(0, 1, 0)
(0, 0, 1)

Part (b)
lex order: $ -3x^5yz^4  + xyz^3 + 2x^2y^8 - xy^4 $
$ LM(f) = x^5yz^4 $
$ LT(f) = - 3x^5yz^4 $
$ multideg(f) = (4, 1, 5) $
(4, 1, 5)
(3, 1, 1)
(0, 8, 2)
(0, 4, 1)

grlex order: $ -3x^5yz^4 + 2x^2y^8 + xyz^3 - xy^4 $
$ LM(f) = x^5yz^4 $
$ LT(f) = - 3x^5yz^4 $
$ multideg(f) = (4, 1, 5) $
(4, 1, 5)
(0, 8, 2)
(3, 1, 1)
(0, 4, 1)

grevlex order: $ 2x^2y^8 - 3x^5yz^4 + xyz^3 - xy^4 $
$ LM(f) = x^2y^8 $
$ LT(f) =  2x^2y^8 $
$ multideg(f) = (0, 8, 2) $
(0, 8, 2)
(4, 1, 5)
(3, 1, 1)
(0, 4, 1)