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"""Tests for Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. """ |
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from sympy.polys.rings import ring |
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from sympy.polys.domains import ZZ, QQ, RR |
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from sympy.polys.specialpolys import ( |
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f_polys, |
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dmp_fateman_poly_F_1, |
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dmp_fateman_poly_F_2, |
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dmp_fateman_poly_F_3) |
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f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys() |
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def test_dup_gcdex(): |
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R, x = ring("x", QQ) |
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f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15 |
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g = x**3 + x**2 - 4*x - 4 |
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s = -QQ(1,5)*x + QQ(3,5) |
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t = QQ(1,5)*x**2 - QQ(6,5)*x + 2 |
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h = x + 1 |
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assert R.dup_half_gcdex(f, g) == (s, h) |
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assert R.dup_gcdex(f, g) == (s, t, h) |
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f = x**4 + 4*x**3 - x + 1 |
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g = x**3 - x + 1 |
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s, t, h = R.dup_gcdex(f, g) |
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S, T, H = R.dup_gcdex(g, f) |
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assert R.dup_add(R.dup_mul(s, f), |
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R.dup_mul(t, g)) == h |
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assert R.dup_add(R.dup_mul(S, g), |
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R.dup_mul(T, f)) == H |
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f = 2*x |
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g = x**2 - 16 |
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s = QQ(1,32)*x |
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t = -QQ(1,16) |
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h = 1 |
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assert R.dup_half_gcdex(f, g) == (s, h) |
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assert R.dup_gcdex(f, g) == (s, t, h) |
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def test_dup_invert(): |
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R, x = ring("x", QQ) |
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assert R.dup_invert(2*x, x**2 - 16) == QQ(1,32)*x |
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def test_dup_euclidean_prs(): |
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R, x = ring("x", QQ) |
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f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 |
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g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 |
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assert R.dup_euclidean_prs(f, g) == [ |
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f, |
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g, |
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-QQ(5,9)*x**4 + QQ(1,9)*x**2 - QQ(1,3), |
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-QQ(117,25)*x**2 - 9*x + QQ(441,25), |
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QQ(233150,19773)*x - QQ(102500,6591), |
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-QQ(1288744821,543589225)] |
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def test_dup_primitive_prs(): |
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R, x = ring("x", ZZ) |
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f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 |
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g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 |
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assert R.dup_primitive_prs(f, g) == [ |
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f, |
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g, |
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-5*x**4 + x**2 - 3, |
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13*x**2 + 25*x - 49, |
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4663*x - 6150, |
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1] |
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def test_dup_subresultants(): |
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R, x = ring("x", ZZ) |
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assert R.dup_resultant(0, 0) == 0 |
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assert R.dup_resultant(1, 0) == 0 |
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assert R.dup_resultant(0, 1) == 0 |
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f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 |
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g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 |
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a = 15*x**4 - 3*x**2 + 9 |
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b = 65*x**2 + 125*x - 245 |
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c = 9326*x - 12300 |
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d = 260708 |
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assert R.dup_subresultants(f, g) == [f, g, a, b, c, d] |
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assert R.dup_resultant(f, g) == R.dup_LC(d) |
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f = x**2 - 2*x + 1 |
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g = x**2 - 1 |
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a = 2*x - 2 |
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assert R.dup_subresultants(f, g) == [f, g, a] |
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assert R.dup_resultant(f, g) == 0 |
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f = x**2 + 1 |
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g = x**2 - 1 |
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a = -2 |
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assert R.dup_subresultants(f, g) == [f, g, a] |
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assert R.dup_resultant(f, g) == 4 |
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f = x**2 - 1 |
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g = x**3 - x**2 + 2 |
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assert R.dup_resultant(f, g) == 0 |
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f = 3*x**3 - x |
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g = 5*x**2 + 1 |
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assert R.dup_resultant(f, g) == 64 |
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f = x**2 - 2*x + 7 |
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g = x**3 - x + 5 |
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assert R.dup_resultant(f, g) == 265 |
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f = x**3 - 6*x**2 + 11*x - 6 |
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g = x**3 - 15*x**2 + 74*x - 120 |
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assert R.dup_resultant(f, g) == -8640 |
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f = x**3 - 6*x**2 + 11*x - 6 |
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g = x**3 - 10*x**2 + 29*x - 20 |
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assert R.dup_resultant(f, g) == 0 |
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f = x**3 - 1 |
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g = x**3 + 2*x**2 + 2*x - 1 |
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assert R.dup_resultant(f, g) == 16 |
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f = x**8 - 2 |
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g = x - 1 |
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assert R.dup_resultant(f, g) == -1 |
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def test_dmp_subresultants(): |
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R, x, y = ring("x,y", ZZ) |
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assert R.dmp_resultant(0, 0) == 0 |
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assert R.dmp_prs_resultant(0, 0)[0] == 0 |
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assert R.dmp_zz_collins_resultant(0, 0) == 0 |
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assert R.dmp_qq_collins_resultant(0, 0) == 0 |
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assert R.dmp_resultant(1, 0) == 0 |
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assert R.dmp_resultant(1, 0) == 0 |
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assert R.dmp_resultant(1, 0) == 0 |
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assert R.dmp_resultant(0, 1) == 0 |
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assert R.dmp_prs_resultant(0, 1)[0] == 0 |
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assert R.dmp_zz_collins_resultant(0, 1) == 0 |
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assert R.dmp_qq_collins_resultant(0, 1) == 0 |
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f = 3*x**2*y - y**3 - 4 |
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g = x**2 + x*y**3 - 9 |
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a = 3*x*y**4 + y**3 - 27*y + 4 |
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b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 |
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r = R.dmp_LC(b) |
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assert R.dmp_subresultants(f, g) == [f, g, a, b] |
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assert R.dmp_resultant(f, g) == r |
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assert R.dmp_prs_resultant(f, g)[0] == r |
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assert R.dmp_zz_collins_resultant(f, g) == r |
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assert R.dmp_qq_collins_resultant(f, g) == r |
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f = -x**3 + 5 |
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g = 3*x**2*y + x**2 |
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a = 45*y**2 + 30*y + 5 |
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b = 675*y**3 + 675*y**2 + 225*y + 25 |
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r = R.dmp_LC(b) |
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assert R.dmp_subresultants(f, g) == [f, g, a] |
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assert R.dmp_resultant(f, g) == r |
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assert R.dmp_prs_resultant(f, g)[0] == r |
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assert R.dmp_zz_collins_resultant(f, g) == r |
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assert R.dmp_qq_collins_resultant(f, g) == r |
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R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) |
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f = 6*x**2 - 3*x*y - 2*x*z + y*z |
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g = x**2 - x*u - x*v + u*v |
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r = y**2*z**2 - 3*y**2*z*u - 3*y**2*z*v + 9*y**2*u*v - 2*y*z**2*u \ |
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- 2*y*z**2*v + 6*y*z*u**2 + 12*y*z*u*v + 6*y*z*v**2 - 18*y*u**2*v \ |
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- 18*y*u*v**2 + 4*z**2*u*v - 12*z*u**2*v - 12*z*u*v**2 + 36*u**2*v**2 |
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assert R.dmp_zz_collins_resultant(f, g) == r.drop(x) |
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R, x, y, z, u, v = ring("x,y,z,u,v", QQ) |
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f = x**2 - QQ(1,2)*x*y - QQ(1,3)*x*z + QQ(1,6)*y*z |
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g = x**2 - x*u - x*v + u*v |
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r = QQ(1,36)*y**2*z**2 - QQ(1,12)*y**2*z*u - QQ(1,12)*y**2*z*v + QQ(1,4)*y**2*u*v \ |
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- QQ(1,18)*y*z**2*u - QQ(1,18)*y*z**2*v + QQ(1,6)*y*z*u**2 + QQ(1,3)*y*z*u*v \ |
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+ QQ(1,6)*y*z*v**2 - QQ(1,2)*y*u**2*v - QQ(1,2)*y*u*v**2 + QQ(1,9)*z**2*u*v \ |
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- QQ(1,3)*z*u**2*v - QQ(1,3)*z*u*v**2 + u**2*v**2 |
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assert R.dmp_qq_collins_resultant(f, g) == r.drop(x) |
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Rt, t = ring("t", ZZ) |
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Rx, x = ring("x", Rt) |
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f = x**6 - 5*x**4 + 5*x**2 + 4 |
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g = -6*t*x**5 + x**4 + 20*t*x**3 - 3*x**2 - 10*t*x + 6 |
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assert Rx.dup_resultant(f, g) == 2930944*t**6 + 2198208*t**4 + 549552*t**2 + 45796 |
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def test_dup_discriminant(): |
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R, x = ring("x", ZZ) |
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assert R.dup_discriminant(0) == 0 |
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assert R.dup_discriminant(x) == 1 |
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assert R.dup_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664 |
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assert R.dup_discriminant(5*x**5 + x**3 + 2) == 31252160 |
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assert R.dup_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0 |
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assert R.dup_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112 |
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def test_dmp_discriminant(): |
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R, x = ring("x", ZZ) |
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assert R.dmp_discriminant(0) == 0 |
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R, x, y = ring("x,y", ZZ) |
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assert R.dmp_discriminant(0) == 0 |
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assert R.dmp_discriminant(y) == 0 |
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assert R.dmp_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664 |
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assert R.dmp_discriminant(5*x**5 + x**3 + 2) == 31252160 |
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assert R.dmp_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0 |
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assert R.dmp_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112 |
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assert R.dmp_discriminant(x**2*y + 2*y) == (-8*y**2).drop(x) |
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assert R.dmp_discriminant(x*y**2 + 2*x) == 1 |
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R, x, y, z = ring("x,y,z", ZZ) |
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assert R.dmp_discriminant(x*y + z) == 1 |
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R, x, y, z, u = ring("x,y,z,u", ZZ) |
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assert R.dmp_discriminant(x**2*y + x*z + u) == (-4*y*u + z**2).drop(x) |
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R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) |
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assert R.dmp_discriminant(x**3*y + x**2*z + x*u + v) == \ |
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(-27*y**2*v**2 + 18*y*z*u*v - 4*y*u**3 - 4*z**3*v + z**2*u**2).drop(x) |
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def test_dup_gcd(): |
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R, x = ring("x", ZZ) |
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f, g = 0, 0 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (0, 0, 0) |
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f, g = 2, 0 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 0) |
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f, g = -2, 0 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 0) |
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f, g = 0, -2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 0, -1) |
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f, g = 0, 2*x + 4 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 0, 1) |
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f, g = 2*x + 4, 0 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 1, 0) |
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f, g = 2, 2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 1) |
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f, g = -2, 2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 1) |
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f, g = 2, -2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, -1) |
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f, g = -2, -2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, -1) |
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f, g = x**2 + 2*x + 1, 1 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1) |
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f, g = x**2 + 2*x + 1, 2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2) |
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f, g = 2*x**2 + 4*x + 2, 2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1) |
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f, g = 2, 2*x**2 + 4*x + 2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1) |
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f, g = 2*x**2 + 4*x + 2, x + 1 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1) |
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f, g = x + 1, 2*x**2 + 4*x + 2 |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2) |
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f, g = x - 31, x |
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assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, f, g) |
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f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 |
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g = x**3 + 6*x**2 + 11*x + 6 |
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h = x**2 + 3*x + 2 |
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cff = x**2 + 5*x + 4 |
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cfg = x + 3 |
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assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) |
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assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg) |
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f = x**4 - 4 |
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g = x**4 + 4*x**2 + 4 |
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h = x**2 + 2 |
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cff = x**2 - 2 |
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cfg = x**2 + 2 |
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assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) |
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assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg) |
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f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 |
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g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 |
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h = 1 |
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cff = f |
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cfg = g |
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assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) |
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assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg) |
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R, x = ring("x", QQ) |
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f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 |
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g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 |
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h = 1 |
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cff = f |
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cfg = g |
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assert R.dup_qq_heu_gcd(f, g) == (h, cff, cfg) |
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assert R.dup_ff_prs_gcd(f, g) == (h, cff, cfg) |
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R, x = ring("x", ZZ) |
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f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \ |
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+ 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \ |
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+ 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \ |
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+ 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \ |
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- 12278371209708240950316872681744825481125965781519138077173235712*x**21 \ |
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+ 289127344604779611146960547954288113529690984687482920704*x**14 \ |
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+ 19007977035740498977629742919480623972236450681*x**7 \ |
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+ 311973482284542371301330321821976049 |
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g = 365431878023781158602430064717380211405897160759702125019136*x**21 \ |
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+ 197599133478719444145775798221171663643171734081650688*x**14 \ |
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- 9504116979659010018253915765478924103928886144*x**7 \ |
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- 311973482284542371301330321821976049 |
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assert R.dup_zz_heu_gcd(f, R.dup_diff(f, 1))[0] == g |
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assert R.dup_rr_prs_gcd(f, R.dup_diff(f, 1))[0] == g |
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R, x = ring("x", QQ) |
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f = QQ(1,2)*x**2 + x + QQ(1,2) |
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g = QQ(1,2)*x + QQ(1,2) |
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h = x + 1 |
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assert R.dup_qq_heu_gcd(f, g) == (h, g, QQ(1,2)) |
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assert R.dup_ff_prs_gcd(f, g) == (h, g, QQ(1,2)) |
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R, x = ring("x", ZZ) |
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f = 1317378933230047068160*x + 2945748836994210856960 |
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g = 120352542776360960*x + 269116466014453760 |
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h = 120352542776360960*x + 269116466014453760 |
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cff = 10946 |
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cfg = 1 |
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assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg) |
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def test_dmp_gcd(): |
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R, x, y = ring("x,y", ZZ) |
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f, g = 0, 0 |
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assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0) |
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f, g = 2, 0 |
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assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0) |
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f, g = -2, 0 |
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assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0) |
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f, g = 0, -2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1) |
|
|
|
f, g = 0, 2*x + 4 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 0, 1) |
|
|
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f, g = 2*x + 4, 0 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 1, 0) |
|
|
|
f, g = 2, 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1) |
|
|
|
f, g = -2, 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1) |
|
|
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f, g = 2, -2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1) |
|
|
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f, g = -2, -2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1) |
|
|
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f, g = x**2 + 2*x + 1, 1 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1) |
|
|
|
f, g = x**2 + 2*x + 1, 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2) |
|
|
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f, g = 2*x**2 + 4*x + 2, 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1) |
|
|
|
f, g = 2, 2*x**2 + 4*x + 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1) |
|
|
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f, g = 2*x**2 + 4*x + 2, x + 1 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1) |
|
|
|
f, g = x + 1, 2*x**2 + 4*x + 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2) |
|
|
|
R, x, y, z, u = ring("x,y,z,u", ZZ) |
|
|
|
f, g = u**2 + 2*u + 1, 2*u + 2 |
|
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (u + 1, u + 1, 2) |
|
|
|
f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1 |
|
h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1 |
|
|
|
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg) |
|
assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg) |
|
|
|
assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff) |
|
assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff) |
|
|
|
R, x, y, z = ring("x,y,z", ZZ) |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ)) |
|
H, cff, cfg = R.dmp_zz_heu_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
H, cff, cfg = R.dmp_rr_prs_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ)) |
|
H, cff, cfg = R.dmp_zz_heu_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ) |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ)) |
|
H, cff, cfg = R.dmp_zz_heu_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ) |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ)) |
|
H, cff, cfg = R.dmp_zz_heu_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
R, x, y, z = ring("x,y,z", ZZ) |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ)) |
|
H, cff, cfg = R.dmp_zz_heu_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
H, cff, cfg = R.dmp_rr_prs_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ)) |
|
H, cff, cfg = R.dmp_zz_heu_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
H, cff, cfg = R.dmp_rr_prs_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) |
|
|
|
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ)) |
|
H, cff, cfg = R.dmp_inner_gcd(f, g) |
|
|
|
assert H == h and R.dmp_mul(H, cff) == f \ |
|
and R.dmp_mul(H, cfg) == g |
|
|
|
R, x, y = ring("x,y", QQ) |
|
|
|
f = QQ(1,2)*x**2 + x + QQ(1,2) |
|
g = QQ(1,2)*x + QQ(1,2) |
|
|
|
h = x + 1 |
|
|
|
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1,2)) |
|
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1,2)) |
|
|
|
R, x, y = ring("x,y", RR) |
|
|
|
f = 2.1*x*y**2 - 2.2*x*y + 2.1*x |
|
g = 1.0*x**3 |
|
|
|
assert R.dmp_ff_prs_gcd(f, g) == \ |
|
(1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2) |
|
|
|
|
|
def test_dup_lcm(): |
|
R, x = ring("x", ZZ) |
|
|
|
assert R.dup_lcm(2, 6) == 6 |
|
|
|
assert R.dup_lcm(2*x**3, 6*x) == 6*x**3 |
|
assert R.dup_lcm(2*x**3, 3*x) == 6*x**3 |
|
|
|
assert R.dup_lcm(x**2 + x, x) == x**2 + x |
|
assert R.dup_lcm(x**2 + x, 2*x) == 2*x**2 + 2*x |
|
assert R.dup_lcm(x**2 + 2*x, x) == x**2 + 2*x |
|
assert R.dup_lcm(2*x**2 + x, x) == 2*x**2 + x |
|
assert R.dup_lcm(2*x**2 + x, 2*x) == 4*x**2 + 2*x |
|
|
|
|
|
def test_dmp_lcm(): |
|
R, x, y = ring("x,y", ZZ) |
|
|
|
assert R.dmp_lcm(2, 6) == 6 |
|
assert R.dmp_lcm(x, y) == x*y |
|
|
|
assert R.dmp_lcm(2*x**3, 6*x*y**2) == 6*x**3*y**2 |
|
assert R.dmp_lcm(2*x**3, 3*x*y**2) == 6*x**3*y**2 |
|
|
|
assert R.dmp_lcm(x**2*y, x*y**2) == x**2*y**2 |
|
|
|
f = 2*x*y**5 - 3*x*y**4 - 2*x*y**3 + 3*x*y**2 |
|
g = y**5 - 2*y**3 + y |
|
h = 2*x*y**7 - 3*x*y**6 - 4*x*y**5 + 6*x*y**4 + 2*x*y**3 - 3*x*y**2 |
|
|
|
assert R.dmp_lcm(f, g) == h |
|
|
|
f = x**3 - 3*x**2*y - 9*x*y**2 - 5*y**3 |
|
g = x**4 + 6*x**3*y + 12*x**2*y**2 + 10*x*y**3 + 3*y**4 |
|
h = x**5 + x**4*y - 18*x**3*y**2 - 50*x**2*y**3 - 47*x*y**4 - 15*y**5 |
|
|
|
assert R.dmp_lcm(f, g) == h |
|
|
|
|
|
def test_dmp_content(): |
|
R, x,y = ring("x,y", ZZ) |
|
|
|
assert R.dmp_content(-2) == 2 |
|
|
|
f, g, F = 3*y**2 + 2*y + 1, 1, 0 |
|
|
|
for i in range(0, 5): |
|
g *= f |
|
F += x**i*g |
|
|
|
assert R.dmp_content(F) == f.drop(x) |
|
|
|
R, x,y,z = ring("x,y,z", ZZ) |
|
|
|
assert R.dmp_content(f_4) == 1 |
|
assert R.dmp_content(f_5) == 1 |
|
|
|
R, x,y,z,t = ring("x,y,z,t", ZZ) |
|
assert R.dmp_content(f_6) == 1 |
|
|
|
|
|
def test_dmp_primitive(): |
|
R, x,y = ring("x,y", ZZ) |
|
|
|
assert R.dmp_primitive(0) == (0, 0) |
|
assert R.dmp_primitive(1) == (1, 1) |
|
|
|
f, g, F = 3*y**2 + 2*y + 1, 1, 0 |
|
|
|
for i in range(0, 5): |
|
g *= f |
|
F += x**i*g |
|
|
|
assert R.dmp_primitive(F) == (f.drop(x), F / f) |
|
|
|
R, x,y,z = ring("x,y,z", ZZ) |
|
|
|
cont, f = R.dmp_primitive(f_4) |
|
assert cont == 1 and f == f_4 |
|
cont, f = R.dmp_primitive(f_5) |
|
assert cont == 1 and f == f_5 |
|
|
|
R, x,y,z,t = ring("x,y,z,t", ZZ) |
|
|
|
cont, f = R.dmp_primitive(f_6) |
|
assert cont == 1 and f == f_6 |
|
|
|
|
|
def test_dup_cancel(): |
|
R, x = ring("x", ZZ) |
|
|
|
f = 2*x**2 - 2 |
|
g = x**2 - 2*x + 1 |
|
|
|
p = 2*x + 2 |
|
q = x - 1 |
|
|
|
assert R.dup_cancel(f, g) == (p, q) |
|
assert R.dup_cancel(f, g, include=False) == (1, 1, p, q) |
|
|
|
f = -x - 2 |
|
g = 3*x - 4 |
|
|
|
F = x + 2 |
|
G = -3*x + 4 |
|
|
|
assert R.dup_cancel(f, g) == (f, g) |
|
assert R.dup_cancel(F, G) == (f, g) |
|
|
|
assert R.dup_cancel(0, 0) == (0, 0) |
|
assert R.dup_cancel(0, 0, include=False) == (1, 1, 0, 0) |
|
|
|
assert R.dup_cancel(x, 0) == (1, 0) |
|
assert R.dup_cancel(x, 0, include=False) == (1, 1, 1, 0) |
|
|
|
assert R.dup_cancel(0, x) == (0, 1) |
|
assert R.dup_cancel(0, x, include=False) == (1, 1, 0, 1) |
|
|
|
f = 0 |
|
g = x |
|
one = 1 |
|
|
|
assert R.dup_cancel(f, g, include=True) == (f, one) |
|
|
|
|
|
def test_dmp_cancel(): |
|
R, x, y = ring("x,y", ZZ) |
|
|
|
f = 2*x**2 - 2 |
|
g = x**2 - 2*x + 1 |
|
|
|
p = 2*x + 2 |
|
q = x - 1 |
|
|
|
assert R.dmp_cancel(f, g) == (p, q) |
|
assert R.dmp_cancel(f, g, include=False) == (1, 1, p, q) |
|
|
|
assert R.dmp_cancel(0, 0) == (0, 0) |
|
assert R.dmp_cancel(0, 0, include=False) == (1, 1, 0, 0) |
|
|
|
assert R.dmp_cancel(y, 0) == (1, 0) |
|
assert R.dmp_cancel(y, 0, include=False) == (1, 1, 1, 0) |
|
|
|
assert R.dmp_cancel(0, y) == (0, 1) |
|
assert R.dmp_cancel(0, y, include=False) == (1, 1, 0, 1) |
|
|