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

The Division Algorithm x2 -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

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 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

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

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 -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+ ) !