From 04af4ba818603b6147f3914202d30b7054a9b507 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 20 Feb 2019 00:38:26 -0500 Subject: [PATCH] mjo/polynomial.py: add two more examples from exercises in the text. --- mjo/polynomial.py | 39 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 39 insertions(+) diff --git a/mjo/polynomial.py b/mjo/polynomial.py index 7414bdf..4fe4b71 100644 --- a/mjo/polynomial.py +++ b/mjo/polynomial.py @@ -75,6 +75,45 @@ def multidiv(f, gs): ....: 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 + TESTS: If we get a zero remainder, then the numerator should belong to -- 2.43.2