4)

R.<x,y> = PolynomialRing(QQ, order = 'lex')
f = x^3*y^3 + 1
p = 1 + 3*x + 2*x^2 + x^2*y + x^3*y
q = x*y^2 + x^2*y^2
print(f)
print(p)
print(q)

x^3*y^3 + 1
x^3*y + x^2*y + 2*x^2 + 3*x + 1
x^2*y^2 + x*y^2

g1 = f - y^2*p
g1

-x^2*y^3 - 2*x^2*y^2 - 3*x*y^2 - y^2 + 1

g2 = g1 + y*q
g2

-2*x^2*y^2 + x*y^3 - 3*x*y^2 - y^2 + 1

r = g2 + 2*q
r

x*y^3 - x*y^2 - y^2 + 1

a1 = y^2
a2 = -y - 2

f == a1*p + a2*q + r

True

# degrees of the leading terms of f, a1*p and a2*q
print(f.lt().degrees())
print((a1*p).lt().degrees())
print((a2*q).lt().degrees())

(3, 3)
(3, 3)
(2, 3)

# faster way (different remainder)
f.reduce([p,q])

x*y^3 + 1

5)

R.<x,y> = PolynomialRing(QQ, order = 'lex')
f = x^3*y^2
p = 1 + x^3*y + 3*x^2*y^5
q = 2*x^2*y + x^2*y^2
print(f)
print(p)
print(q)

x^3*y^2
x^3*y + 3*x^2*y^5 + 1
x^2*y^2 + 2*x^2*y

g1 = f - y*p
g1

-3*x^2*y^6 - y

g2 = g1 + 3*y^4*q
g2

6*x^2*y^5 - y

g3 = g2 - 6*y^3*q
g3

-12*x^2*y^4 - y

g4 = g3 + 12*y^2*q
g4

24*x^2*y^3 - y

g5 = g4 - 24*y*q
g5

-48*x^2*y^2 - y

r = g5 + 48*q
r

96*x^2*y - y

a1 = y
a2 = -3*y^4 + 6*y^3 - 12*y^2 + 24*y - 48

f == a1*p + a2*q + r

True

# degrees of the leading terms of f, a1*p and a2*q
print(f.lt().degrees())
print((a1*p).lt().degrees())
print((a2*q).lt().degrees())

(3, 2)
(3, 2)
(2, 6)

# faster way
f.reduce([p,q])

96*x^2*y + 2

6)

R.<x,y> = PolynomialRing(QQ, order = 'lex')
f = x^2*y + x*y^2 + y^2
f1 = x*y - 1
f2 = y^2 - 1
print(f)
print(f1)
print(f2)

x^2*y + x*y^2 + y^2
x*y - 1
y^2 - 1

# reduce first by f1 and then by f2
r = f.reduce([f1]).reduce([f2])
r

x + y + 1

a1 = -x - y
a2 = -1

f + a1*f1 + a2*f2

x + y + 1

# reduce by f1 and f2
r = f.reduce([f1, f2])
r

2*x + 1

# different reduction with b1, b2
b1 = -x
b2 = -1 - x

f + b1*f1 + b2*f2

2*x + 1