UTM Ideals Varieties and Algorithm - Chapter 2 Section 2 Exercise 1

Problem:

Solution:

It is a really simple sorting exercise, so let's do some warmup.

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

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

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

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

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

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