Multivariate polynomial division using Sage

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I'm just starting to learn Sage and I'm trying to find some method to do polynomial divisions in several variables. An example of the calculations that I want to do can be the division of the polynomial f(x)=x^2*y+x*y^2+y^2 with the polynomials f1(x)=x*y-1 and f2( x)=y^2-1 using lexicographic order. The result should be f(x)=(x+y)*(x*y-1)+1*(y^2-1)+(x+y+1). My interest would be to have a method in Sage to find the quotients, (x+y), (1), and the remainder, (x+y+1). Sage has many functions for similar and more complex operations, like reduce, quo_rem or groebner_basis(), but I haven't found one that does what I need.

Thanks for your help.

sage division
1 Answers

A way to obtain the result in the given sample case is as follows. Introduce the ring R = Q[x,y], and inside it build the ideal J generated by the two polynomials f1 and f2. Then the "rest" above will be a representation of f in the quotient ring, R/J. (Ring modulo ideal.) This rest can be lifted from the quotient ring to an element r of R. Then the difference f - r is an element of the ideal, and "lifting" this element is giving the two components to be multiplied with f1 and f2. Let's see:

R.<x,y> = PolynomialRing(QQ)
f = x^2*y + x*y^2 + y^2
f1 = x*y - 1
f2 = y^2 - 1
J = R.ideal([f1, f2])

Q = R.quotient(J)
r = Q(f).lift()

q1, q2 = (f - r).lift(J)
print(f'r = {r}')
print(f'q1 = {q1}')
print(f'q2 = {q2}')

And the above code delivers:

r = 2*y + 1
q1 = x*y^2 + 2*y
q2 = -x^2*y + 1

The question expects an r equal to x + y + 1, but observe that the difference x - y is in the ideal J, which has the following generators of the...

sage: J.groebner_basis()
[y^2 - 1, x - y]
user_102425950
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