sage: x,y = R.gens()
sage: f = x*y^2 + 1
sage: gs = [ x*y + 1, y + 1 ]
- sage: multidiv(f, gs)
+ sage: (qs, r) = multidiv(f, gs)
+ sage: (qs, r)
([y, -1], 2)
+ sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+ True
+ sage: not any( g.lt().divides(m) for m in r.monomials()
+ ....: for g in gs )
+ True
Example 2 in Section 2.3 of Cox, Little, and O'Shea::
sage: x,y = R.gens()
sage: f = x^2*y + x*y^2 + y^2
sage: gs = [ x*y - 1, y^2 - 1 ]
- sage: multidiv(f, gs)
+ sage: (qs, r) = multidiv(f, gs)
+ sage: (qs, r)
([x + y, 1], x + y + 1)
+ sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+ True
+ sage: not any( g.lt().divides(m) for m in r.monomials()
+ ....: for g in gs )
+ True
+
+ A solution ``g`` to Exercise 6 in Section 2.3 of Cox, Little, and
+ O'Shea that lives in the ideal generated by ``f1`` and ``f2`` but
+ which has nonzero remainder after division::
+
+ sage: R = PolynomialRing(QQ, 'x,y', order='deglex')
+ sage: x,y = R.gens()
+ sage: f1 = 2*x*y^2 - x
+ sage: f2 = 3*x^2*y - y - 1
+ sage: I = R.ideal(f1,f2)
+ sage: g = 2*y*f2
+ sage: g in I
+ True
+ sage: (qs,r) = multidiv(g,[f1,f2])
+ sage: r.is_zero()
+ False
+
+ Two solutions ``g`` to Exercise 7 in Section 2.3 of Cox, Little, and
+ O'Shea that live in the ideal generated by ``f1``, ``f2``, and ``f3``
+ but which have nonzero remainders after division::
+
+ sage: R = PolynomialRing(QQ, 'x,y,z', order='deglex')
+ sage: x,y,z = R.gens()
+ sage: f1 = x^4*y^2 - z
+ sage: f2 = x^3*y^3 - 1
+ sage: f3 = x^2*y^4 - 2*z
+ sage: I = R.ideal(f1,f2,f3)
+ sage: g = x^2*f3
+ sage: g in I
+ True
+ sage: (qs, r) = multidiv(g, [f1,f2,f3])
+ sage: r.is_zero()
+ False
+ sage: g = x*f2
+ sage: g in I
+ True
+ sage: (qs, r) = multidiv(g, [f1,f2,f3])
+ sage: r.is_zero()
+ False
+
+ Example 4 in Section 2.3 of Cox, Little, and O'Shea. This is the
+ same as Example 2, except with the order of ``gs`` reversed::
+
+ sage: R = PolynomialRing(QQ, 'x,y', order='lex')
+ sage: x,y = R.gens()
+ sage: f = x^2*y + x*y^2 + y^2
+ sage: gs = [ y^2 - 1, x*y - 1 ]
+ sage: (qs, r) = multidiv(f, gs)
+ sage: (qs, r)
+ ([x + 1, x], 2*x + 1)
+ sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+ True
+ sage: not any( g.lt().divides(m) for m in r.monomials()
+ ....: for g in gs )
+ True
TESTS:
- Derp.
+ If we get a zero remainder, then the numerator should belong to
+ the ideal generated by the denominators::
+
+ sage: set_random_seed()
+ sage: R = PolynomialRing(QQ, 'x,y,z')
+ sage: x,y,z = R.gens()
+ sage: s = ZZ.random_element(1,5).abs()
+ sage: gs = [ R.random_element() for idx in range(s) ]
+ sage: # hack for SageMath Trac #28855
+ sage: f = R(R.random_element(ZZ.random_element(10).abs()))
+ sage: (qs, r) = multidiv(f,gs)
+ sage: r != 0 or f in R.ideal(gs)
+ True
+
+ The numerator is always the sum of the remainder and the quotients
+ times the denominators, and the remainder's monomials aren't divisible
+ by the leading term of any denominator::
+
+ sage: set_random_seed()
+ sage: R = PolynomialRing(QQ, 'x,y,z')
+ sage: s = ZZ.random_element(1,5).abs()
+ sage: gs = [ R.random_element() for idx in range(s) ]
+ sage: # hack for SageMath Trac #28855
+ sage: f = R(R.random_element(ZZ.random_element(10).abs()))
+ sage: (qs, r) = multidiv(f,gs)
+ sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+ True
+ sage: r == 0 or (not any( g.lt().divides(m) for m in r.monomials()
+ ....: for g in gs ))
+ True
+
+ Exercise 8 in Section 2.4 of Cox, Little, and O'Shea says that we
+ should always get a zero remainder if we divide an element of a
+ monomial ideal by its generators::
+
+ sage: set_random_seed()
+ sage: R = PolynomialRing(QQ,'x,y,z')
+ sage: gs = R.random_element().monomials()
+ sage: I = R.ideal(gs)
+ sage: # hack for SageMath Trac #28855
+ sage: f = R(I.random_element(ZZ.random_element(5).abs()))
+ sage: (qs, r) = multidiv(f, gs)
+ sage: r.is_zero()
+ True
"""
R = f.parent()