Fun with Polynomials

1. x2 x3 Fun with Polynomials y-3x5 -1+2y3 3 1-6x-y13 3y3 6x-2xyz+5z 5x2 + xy3-6xy 10-6x-x10 Applying the one-variable polynomial division algorithm to several variables -4z2-3xz -x5 + 4yz 5xy+5x2 -10x - 4y7 16x-20xz+5z 16x-200xyz+5z -4z2-3xz -xy5 + 4yz

2. The Division Algorithm x2 -3x+10 1 3 x+3 x3 + x + 6 5 6 8 5 x3 +3x2 Choose the leading terms 1 8 -3x2 + x +6 Proceed as usual 1 5 -3x2 - 9x 3 10x+6 remainder 10x+30 -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? 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 . 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 x+y 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, 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

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