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| """Test modules.py code.""" | |
| from sympy.polys.agca.modules import FreeModule, ModuleOrder, FreeModulePolyRing | |
| from sympy.polys import CoercionFailed, QQ, lex, grlex, ilex, ZZ | |
| from sympy.abc import x, y, z | |
| from sympy.testing.pytest import raises | |
| from sympy.core.numbers import Rational | |
| def test_FreeModuleElement(): | |
| M = QQ.old_poly_ring(x).free_module(3) | |
| e = M.convert([1, x, x**2]) | |
| f = [QQ.old_poly_ring(x).convert(1), QQ.old_poly_ring(x).convert(x), QQ.old_poly_ring(x).convert(x**2)] | |
| assert list(e) == f | |
| assert f[0] == e[0] | |
| assert f[1] == e[1] | |
| assert f[2] == e[2] | |
| raises(IndexError, lambda: e[3]) | |
| g = M.convert([x, 0, 0]) | |
| assert e + g == M.convert([x + 1, x, x**2]) | |
| assert f + g == M.convert([x + 1, x, x**2]) | |
| assert -e == M.convert([-1, -x, -x**2]) | |
| assert e - g == M.convert([1 - x, x, x**2]) | |
| assert e != g | |
| assert M.convert([x, x, x]) / QQ.old_poly_ring(x).convert(x) == [1, 1, 1] | |
| R = QQ.old_poly_ring(x, order="ilex") | |
| assert R.free_module(1).convert([x]) / R.convert(x) == [1] | |
| def test_FreeModule(): | |
| M1 = FreeModule(QQ.old_poly_ring(x), 2) | |
| assert M1 == FreeModule(QQ.old_poly_ring(x), 2) | |
| assert M1 != FreeModule(QQ.old_poly_ring(y), 2) | |
| assert M1 != FreeModule(QQ.old_poly_ring(x), 3) | |
| M2 = FreeModule(QQ.old_poly_ring(x, order="ilex"), 2) | |
| assert [x, 1] in M1 | |
| assert [x] not in M1 | |
| assert [2, y] not in M1 | |
| assert [1/(x + 1), 2] not in M1 | |
| e = M1.convert([x, x**2 + 1]) | |
| X = QQ.old_poly_ring(x).convert(x) | |
| assert e == [X, X**2 + 1] | |
| assert e == [x, x**2 + 1] | |
| assert 2*e == [2*x, 2*x**2 + 2] | |
| assert e*2 == [2*x, 2*x**2 + 2] | |
| assert e/2 == [x/2, (x**2 + 1)/2] | |
| assert x*e == [x**2, x**3 + x] | |
| assert e*x == [x**2, x**3 + x] | |
| assert X*e == [x**2, x**3 + x] | |
| assert e*X == [x**2, x**3 + x] | |
| assert [x, 1] in M2 | |
| assert [x] not in M2 | |
| assert [2, y] not in M2 | |
| assert [1/(x + 1), 2] in M2 | |
| e = M2.convert([x, x**2 + 1]) | |
| X = QQ.old_poly_ring(x, order="ilex").convert(x) | |
| assert e == [X, X**2 + 1] | |
| assert e == [x, x**2 + 1] | |
| assert 2*e == [2*x, 2*x**2 + 2] | |
| assert e*2 == [2*x, 2*x**2 + 2] | |
| assert e/2 == [x/2, (x**2 + 1)/2] | |
| assert x*e == [x**2, x**3 + x] | |
| assert e*x == [x**2, x**3 + x] | |
| assert e/(1 + x) == [x/(1 + x), (x**2 + 1)/(1 + x)] | |
| assert X*e == [x**2, x**3 + x] | |
| assert e*X == [x**2, x**3 + x] | |
| M3 = FreeModule(QQ.old_poly_ring(x, y), 2) | |
| assert M3.convert(e) == M3.convert([x, x**2 + 1]) | |
| assert not M3.is_submodule(0) | |
| assert not M3.is_zero() | |
| raises(NotImplementedError, lambda: ZZ.old_poly_ring(x).free_module(2)) | |
| raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2)) | |
| raises(CoercionFailed, lambda: M1.convert(QQ.old_poly_ring(x).free_module(3) | |
| .convert([1, 2, 3]))) | |
| raises(CoercionFailed, lambda: M3.convert(1)) | |
| def test_ModuleOrder(): | |
| o1 = ModuleOrder(lex, grlex, False) | |
| o2 = ModuleOrder(ilex, lex, False) | |
| assert o1 == ModuleOrder(lex, grlex, False) | |
| assert (o1 != ModuleOrder(lex, grlex, False)) is False | |
| assert o1 != o2 | |
| assert o1((1, 2, 3)) == (1, (5, (2, 3))) | |
| assert o2((1, 2, 3)) == (-1, (2, 3)) | |
| def test_SubModulePolyRing_global(): | |
| R = QQ.old_poly_ring(x, y) | |
| F = R.free_module(3) | |
| Fd = F.submodule([1, 0, 0], [1, 2, 0], [1, 2, 3]) | |
| M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1]) | |
| assert F == Fd | |
| assert Fd == F | |
| assert F != M | |
| assert M != F | |
| assert Fd != M | |
| assert M != Fd | |
| assert Fd == F.submodule(*F.basis()) | |
| assert Fd.is_full_module() | |
| assert not M.is_full_module() | |
| assert not Fd.is_zero() | |
| assert not M.is_zero() | |
| assert Fd.submodule().is_zero() | |
| assert M.contains([x**2 + y**2 + x, 1 + y, 1]) | |
| assert not M.contains([x**2 + y**2 + x, 1 + y, 2]) | |
| assert M.contains([y**2, 1 - x*y, -x]) | |
| assert not F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0]) | |
| assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F | |
| assert not M.is_submodule(0) | |
| m = F.convert([x**2 + y**2, 1, 0]) | |
| n = M.convert(m) | |
| assert m.module is F | |
| assert n.module is M | |
| raises(ValueError, lambda: M.submodule([1, 0, 0])) | |
| raises(TypeError, lambda: M.union(1)) | |
| raises(ValueError, lambda: M.union(R.free_module(1).submodule([x]))) | |
| assert F.submodule([x, x, x]) != F.submodule([x, x, x], order="ilex") | |
| def test_SubModulePolyRing_local(): | |
| R = QQ.old_poly_ring(x, y, order=ilex) | |
| F = R.free_module(3) | |
| Fd = F.submodule([1 + x, 0, 0], [1 + y, 2 + 2*y, 0], [1, 2, 3]) | |
| M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1]) | |
| assert F == Fd | |
| assert Fd == F | |
| assert F != M | |
| assert M != F | |
| assert Fd != M | |
| assert M != Fd | |
| assert Fd == F.submodule(*F.basis()) | |
| assert Fd.is_full_module() | |
| assert not M.is_full_module() | |
| assert not Fd.is_zero() | |
| assert not M.is_zero() | |
| assert Fd.submodule().is_zero() | |
| assert M.contains([x**2 + y**2 + x, 1 + y, 1]) | |
| assert not M.contains([x**2 + y**2 + x, 1 + y, 2]) | |
| assert M.contains([y**2, 1 - x*y, -x]) | |
| assert F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0]) | |
| assert F.submodule( | |
| [1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1 + x*y])) == F | |
| raises(ValueError, lambda: M.submodule([1, 0, 0])) | |
| def test_SubModulePolyRing_nontriv_global(): | |
| R = QQ.old_poly_ring(x, y, z) | |
| F = R.free_module(1) | |
| def contains(I, f): | |
| return F.submodule(*[[g] for g in I]).contains([f]) | |
| assert contains([x, y], x) | |
| assert contains([x, y], x + y) | |
| assert not contains([x, y], 1) | |
| assert not contains([x, y], z) | |
| assert contains([x**2 + y, x**2 + x], x - y) | |
| assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2) | |
| assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3) | |
| assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4) | |
| assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2) | |
| assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x) | |
| assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z) | |
| assert contains([x, 1 + x + y, 5 - 7*y], 1) | |
| assert contains( | |
| [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z], | |
| x**3) | |
| assert not contains( | |
| [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z], | |
| x**2 + y**2) | |
| # compare local order | |
| assert not contains([x*(1 + x + y), y*(1 + z)], x) | |
| assert not contains([x*(1 + x + y), y*(1 + z)], x + y) | |
| def test_SubModulePolyRing_nontriv_local(): | |
| R = QQ.old_poly_ring(x, y, z, order=ilex) | |
| F = R.free_module(1) | |
| def contains(I, f): | |
| return F.submodule(*[[g] for g in I]).contains([f]) | |
| assert contains([x, y], x) | |
| assert contains([x, y], x + y) | |
| assert not contains([x, y], 1) | |
| assert not contains([x, y], z) | |
| assert contains([x**2 + y, x**2 + x], x - y) | |
| assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2) | |
| assert contains([x*(1 + x + y), y*(1 + z)], x) | |
| assert contains([x*(1 + x + y), y*(1 + z)], x + y) | |
| def test_syzygy(): | |
| R = QQ.old_poly_ring(x, y, z) | |
| M = R.free_module(1).submodule([x*y], [y*z], [x*z]) | |
| S = R.free_module(3).submodule([0, x, -y], [z, -x, 0]) | |
| assert M.syzygy_module() == S | |
| M2 = M / ([x*y*z],) | |
| S2 = R.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y]) | |
| assert M2.syzygy_module() == S2 | |
| F = R.free_module(3) | |
| assert F.submodule(*F.basis()).syzygy_module() == F.submodule() | |
| R2 = QQ.old_poly_ring(x, y, z) / [x*y*z] | |
| M3 = R2.free_module(1).submodule([x*y], [y*z], [x*z]) | |
| S3 = R2.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y]) | |
| assert M3.syzygy_module() == S3 | |
| def test_in_terms_of_generators(): | |
| R = QQ.old_poly_ring(x, order="ilex") | |
| M = R.free_module(2).submodule([2*x, 0], [1, 2]) | |
| assert M.in_terms_of_generators( | |
| [x, x]) == [R.convert(Rational(1, 4)), R.convert(x/2)] | |
| raises(ValueError, lambda: M.in_terms_of_generators([1, 0])) | |
| M = R.free_module(2) / ([x, 0], [1, 1]) | |
| SM = M.submodule([1, x]) | |
| assert SM.in_terms_of_generators([2, 0]) == [R.convert(-2/(x - 1))] | |
| R = QQ.old_poly_ring(x, y) / [x**2 - y**2] | |
| M = R.free_module(2) | |
| SM = M.submodule([x, 0], [0, y]) | |
| assert SM.in_terms_of_generators( | |
| [x**2, x**2]) == [R.convert(x), R.convert(y)] | |
| def test_QuotientModuleElement(): | |
| R = QQ.old_poly_ring(x) | |
| F = R.free_module(3) | |
| N = F.submodule([1, x, x**2]) | |
| M = F/N | |
| e = M.convert([x**2, 2, 0]) | |
| assert M.convert([x + 1, x**2 + x, x**3 + x**2]) == 0 | |
| assert e == [x**2, 2, 0] + N == F.convert([x**2, 2, 0]) + N == \ | |
| M.convert(F.convert([x**2, 2, 0])) | |
| assert M.convert([x**2 + 1, 2*x + 2, x**2]) == e + [0, x, 0] == \ | |
| e + M.convert([0, x, 0]) == e + F.convert([0, x, 0]) | |
| assert M.convert([x**2 + 1, 2, x**2]) == e - [0, x, 0] == \ | |
| e - M.convert([0, x, 0]) == e - F.convert([0, x, 0]) | |
| assert M.convert([0, 2, 0]) == M.convert([x**2, 4, 0]) - e == \ | |
| [x**2, 4, 0] - e == F.convert([x**2, 4, 0]) - e | |
| assert M.convert([x**3 + x**2, 2*x + 2, 0]) == (1 + x)*e == \ | |
| R.convert(1 + x)*e == e*(1 + x) == e*R.convert(1 + x) | |
| assert -e == [-x**2, -2, 0] | |
| f = [x, x, 0] + N | |
| assert M.convert([1, 1, 0]) == f / x == f / R.convert(x) | |
| M2 = F/[(2, 2*x, 2*x**2), (0, 0, 1)] | |
| G = R.free_module(2) | |
| M3 = G/[[1, x]] | |
| M4 = F.submodule([1, x, x**2], [1, 0, 0]) / N | |
| raises(CoercionFailed, lambda: M.convert(G.convert([1, x]))) | |
| raises(CoercionFailed, lambda: M.convert(M3.convert([1, x]))) | |
| raises(CoercionFailed, lambda: M.convert(M2.convert([1, x, x]))) | |
| assert M2.convert(M.convert([2, x, x**2])) == [2, x, 0] | |
| assert M.convert(M4.convert([2, 0, 0])) == [2, 0, 0] | |
| def test_QuotientModule(): | |
| R = QQ.old_poly_ring(x) | |
| F = R.free_module(3) | |
| N = F.submodule([1, x, x**2]) | |
| M = F/N | |
| assert M != F | |
| assert M != N | |
| assert M == F / [(1, x, x**2)] | |
| assert not M.is_zero() | |
| assert (F / F.basis()).is_zero() | |
| SQ = F.submodule([1, x, x**2], [2, 0, 0]) / N | |
| assert SQ == M.submodule([2, x, x**2]) | |
| assert SQ != M.submodule([2, 1, 0]) | |
| assert SQ != M | |
| assert M.is_submodule(SQ) | |
| assert not SQ.is_full_module() | |
| raises(ValueError, lambda: N/F) | |
| raises(ValueError, lambda: F.submodule([2, 0, 0]) / N) | |
| raises(ValueError, lambda: R.free_module(2)/F) | |
| raises(CoercionFailed, lambda: F.convert(M.convert([1, x, x**2]))) | |
| M1 = F / [[1, 1, 1]] | |
| M2 = M1.submodule([1, 0, 0], [0, 1, 0]) | |
| assert M1 == M2 | |
| def test_ModulesQuotientRing(): | |
| R = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1] | |
| M1 = R.free_module(2) | |
| assert M1 == R.free_module(2) | |
| assert M1 != QQ.old_poly_ring(x).free_module(2) | |
| assert M1 != R.free_module(3) | |
| assert [x, 1] in M1 | |
| assert [x] not in M1 | |
| assert [1/(R.convert(x) + 1), 2] in M1 | |
| assert [1, 2/(1 + y)] in M1 | |
| assert [1, 2/y] not in M1 | |
| assert M1.convert([x**2, y]) == [-1, y] | |
| F = R.free_module(3) | |
| Fd = F.submodule([x**2, 0, 0], [1, 2, 0], [1, 2, 3]) | |
| M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1]) | |
| assert F == Fd | |
| assert Fd == F | |
| assert F != M | |
| assert M != F | |
| assert Fd != M | |
| assert M != Fd | |
| assert Fd == F.submodule(*F.basis()) | |
| assert Fd.is_full_module() | |
| assert not M.is_full_module() | |
| assert not Fd.is_zero() | |
| assert not M.is_zero() | |
| assert Fd.submodule().is_zero() | |
| assert M.contains([x**2 + y**2 + x, -x**2 + y, 1]) | |
| assert not M.contains([x**2 + y**2 + x, 1 + y, 2]) | |
| assert M.contains([y**2, 1 - x*y, -x]) | |
| assert F.submodule([x, 0, 0]) == F.submodule([1, 0, 0]) | |
| assert not F.submodule([y, 0, 0]) == F.submodule([1, 0, 0]) | |
| assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F | |
| assert not M.is_submodule(0) | |
| def test_module_mul(): | |
| R = QQ.old_poly_ring(x) | |
| M = R.free_module(2) | |
| S1 = M.submodule([x, 0], [0, x]) | |
| S2 = M.submodule([x**2, 0], [0, x**2]) | |
| I = R.ideal(x) | |
| assert I*M == M*I == S1 == x*M == M*x | |
| assert I*S1 == S2 == x*S1 | |
| def test_intersection(): | |
| # SCA, example 2.8.5 | |
| F = QQ.old_poly_ring(x, y).free_module(2) | |
| M1 = F.submodule([x, y], [y, 1]) | |
| M2 = F.submodule([0, y - 1], [x, 1], [y, x]) | |
| I = F.submodule([x, y], [y**2 - y, y - 1], [x*y + y, x + 1]) | |
| I1, rel1, rel2 = M1.intersect(M2, relations=True) | |
| assert I1 == M2.intersect(M1) == I | |
| for i, g in enumerate(I1.gens): | |
| assert g == sum(c*x for c, x in zip(rel1[i], M1.gens)) \ | |
| == sum(d*y for d, y in zip(rel2[i], M2.gens)) | |
| assert F.submodule([x, y]).intersect(F.submodule([y, x])).is_zero() | |
| def test_quotient(): | |
| # SCA, example 2.8.6 | |
| R = QQ.old_poly_ring(x, y, z) | |
| F = R.free_module(2) | |
| assert F.submodule([x*y, x*z], [y*z, x*y]).module_quotient( | |
| F.submodule([y, z], [z, y])) == QQ.old_poly_ring(x, y, z).ideal(x**2*y**2 - x*y*z**2) | |
| assert F.submodule([x, y]).module_quotient(F.submodule()).is_whole_ring() | |
| M = F.submodule([x**2, x**2], [y**2, y**2]) | |
| N = F.submodule([x + y, x + y]) | |
| q, rel = M.module_quotient(N, relations=True) | |
| assert q == R.ideal(y**2, x - y) | |
| for i, g in enumerate(q.gens): | |
| assert g*N.gens[0] == sum(c*x for c, x in zip(rel[i], M.gens)) | |
| def test_groebner_extendend(): | |
| M = QQ.old_poly_ring(x, y, z).free_module(3).submodule([x + 1, y, 1], [x*y, z, z**2]) | |
| G, R = M._groebner_vec(extended=True) | |
| for i, g in enumerate(G): | |
| assert g == sum(c*gen for c, gen in zip(R[i], M.gens)) | |