1. Fun with Polynomials x2 x3 y-3x5 -1+2y3 3 1-6x-y13 3y3 6x-2xyz+5z
Applying the one-variable polynomial division algorithm to several variables
-4z 2-3xz
-x5 + 4yz
5xy+5x2
-10x - 4y7
16x-20xz+5z
16x-200xyz+5z
-4z 2-3xz
-xy5 + 4yz

2. The Division Algorithmx2
-3x+10 1 3 5
6 8
x+3
x3 + x + 6 5
x3 +3x2
Choose the leading terms 1 8
-3x2 + x +6
Proceed as usual
1 5
-3x2 - 9x 3
remainder
10x+6
10x+30 3
remainder
-24
The answer is 13
remainder 5
divisor
Algorithm terminates when we
get a difference with degree less
than that of the divisor

3. But what about multivariable polynomials?
x+y x2 + 2xy + y2
What is the leading term of x+y? x2+2xy+y2 ?

4. Monomial Orderings
Would like to order the monomials of x2 + 2xy + y2 .
x xy y2
Try ordering by degree
x2 , xy, y2 all have degree two, so need a way to break ties
Give x precedence over y
x2 precedes xy precedes y2

5. Back to our problem x + y x+y x2 + 2xy + y2 Identify leading terms
xy is the leading term here
y2 + xy

6. The ordering goes like this
First, order the variables
Next, order monomials by degree
Lastly, break ties using the order on the variables
For example, let’s order the following monomials
xy y x2y x2y xy3
First, say x precedes y
If we order by degree we have
xy x2y x2y y xy2
After breaking ties using the precedence of x we get
x2y2
xy3
x2y
xy2 y3

7. One last time y2 +xy x2y + 2xy2 - x2y2 + y3 -xy3 -xy + x + y
xy+y2 - x2y2 - xy3 +x2y +2xy2 +y3
Order the monomials
-x2y2 - xy3
x2y +2xy2 +y3
x2y + xy2
xy2 +y3
xy2 +y3
So x2y + 2xy2 - x2y2 + y3 -xy3 equals (xy+y2) (-xy+x+ ) !