Fun with Polynomials

5x 2 + xy 3-6xy

3y3

-4z 2-3xz

1-6x-y13

-x5 + 4yz

-10x - 4y 7

5xy+5x 2

16x-200xyz+5z

16x-20xz+5z

-4z 2-3xz

10-6x-x10

-xy5 + 4yz

6x-2xyz+5z

y-3x5

-1+2y3

x2 x3

3

Applying the one-variable polynomial division algorithm to several variables

The Division Algorithm

5 6 851

1 5

3

1

5

3

8

3

remainder

The answer is 13remainder

divisor

x+3 x3 + x + 6Choose the leading terms

x2

x3 +3x2

Proceed as usual-3x2 + x +6

-3x+10

-3x2 - 9x

10x+6 10x+30

-24 remainder

Algorithm terminates when weget a difference with degree less than that of the divisor

But what about multivariable polynomials?

x+y x2 + 2xy + y2

What is the leading term of x+y? x2+2xy+y2 ?

Monomial Orderings

Would like to order the monomials of x2 + 2xy + y2 . x2 xy y2

Try ordering by degreex2 , xy, y2 all have degree two, so need a way to break ties

Give x precedence over y

x2 precedes xy precedes y2

Back to our problem

x+y x2 + 2xy + y2 Identify leading terms

x

y2 + xy xy is the leading term here

+ y

x2 + xy

y2 + xy

0

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

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

One last time

y2 +xy x2y + 2xy2 - x2y2 + y3 -xy3

Order the monomialsxy+y2 - x2y2 - xy3 +x2y +2xy2 +y3

-xy

-x2y2 - xy3

x2y +2xy2 +y3

+ x

x2y + xy2

xy2 +y3

+ y

xy2 +y3

0

So x2y + 2xy2 - x2y2 + y3 -xy3 equals (xy+y2) (-xy+x+ ) !