Fun with Polynomials

1
Fun with Polynomials
x2
x3
y-3x5
-12y3
3
3y3
1-6x-y13
6x-2xyz5z
5x2 xy3-6xy
10-6x-x10
Applying the one-variable polynomial division
algorithm to several variables
-4z 2-3xz
-x5 4yz
5xy5x2
-10x - 4y7
16x-20xz5z
16x-200xyz5z
-4z 2-3xz
-xy5 4yz
2
The Division Algorithm
x2
-3x10
1
3
x3
x3 x 6
5
6 8
5
x3 3x2
Choose the leading terms
1
8
Proceed as usual
-3x2 x 6
1 5
-3x2 - 9x
3
10x6
remainder
10x30
-24
3
remainder
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?
xy x2 2xy y2
What is the leading term of xy? x22xyy2 ?
4
Monomial Orderings
Would like to order the monomials of x2 2xy
y2 . x2 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
xy x2 2xy y2
Identify leading terms
x2 xy
y2 xy
xy is the leading term here
y2 xy
0
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, lets order the following monomials
xy2 y3 x2y2
x2y xy3

• First, say x precedes y

• If we order by degree we have

xy3 x2y2 x2y
y3 xy2

• After breaking ties using the precedence of x we
  get

x2y2
xy3
x2y
xy2
y3
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One last time
y2 xy x2y 2xy2 - x2y2 y3 -xy3
-xy
x
y
Order the monomials
xyy2 - x2y2 - xy3 x2y 2xy2 y3
-x2y2 - xy3
x2y 2xy2 y3
x2y xy2
xy2 y3
xy2 y3
0
So x2y 2xy2 - x2y2 y3 -xy3 equals (xyy2)
(-xyx ) !