475 lines
22 KiB
Python
475 lines
22 KiB
Python
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"""Tests for minimal polynomials. """
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from sympy.core.function import expand
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from sympy.core import (GoldenRatio, TribonacciConstant)
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from sympy.core.numbers import (AlgebraicNumber, I, Rational, oo, pi)
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from sympy.core.power import Pow
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from sympy.core.singleton import S
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import (cbrt, sqrt)
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from sympy.functions.elementary.trigonometric import (cos, sin, tan)
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from sympy.polys.polytools import Poly
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from sympy.polys.rootoftools import CRootOf
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from sympy.solvers.solveset import nonlinsolve
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from sympy.geometry import Circle, intersection
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from sympy.testing.pytest import raises, slow
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from sympy.sets.sets import FiniteSet
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from sympy.geometry.point import Point2D
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from sympy.polys.numberfields.minpoly import (
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minimal_polynomial,
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_choose_factor,
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_minpoly_op_algebraic_element,
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_separate_sq,
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_minpoly_groebner,
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)
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from sympy.polys.partfrac import apart
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from sympy.polys.polyerrors import (
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NotAlgebraic,
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GeneratorsError,
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)
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from sympy.polys.domains import QQ
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from sympy.polys.rootoftools import rootof
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from sympy.polys.polytools import degree
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from sympy.abc import x, y, z
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Q = Rational
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def test_minimal_polynomial():
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assert minimal_polynomial(-7, x) == x + 7
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assert minimal_polynomial(-1, x) == x + 1
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assert minimal_polynomial( 0, x) == x
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assert minimal_polynomial( 1, x) == x - 1
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assert minimal_polynomial( 7, x) == x - 7
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assert minimal_polynomial(sqrt(2), x) == x**2 - 2
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assert minimal_polynomial(sqrt(5), x) == x**2 - 5
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assert minimal_polynomial(sqrt(6), x) == x**2 - 6
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assert minimal_polynomial(2*sqrt(2), x) == x**2 - 8
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assert minimal_polynomial(3*sqrt(5), x) == x**2 - 45
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assert minimal_polynomial(4*sqrt(6), x) == x**2 - 96
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assert minimal_polynomial(2*sqrt(2) + 3, x) == x**2 - 6*x + 1
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assert minimal_polynomial(3*sqrt(5) + 6, x) == x**2 - 12*x - 9
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assert minimal_polynomial(4*sqrt(6) + 7, x) == x**2 - 14*x - 47
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assert minimal_polynomial(2*sqrt(2) - 3, x) == x**2 + 6*x + 1
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assert minimal_polynomial(3*sqrt(5) - 6, x) == x**2 + 12*x - 9
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assert minimal_polynomial(4*sqrt(6) - 7, x) == x**2 + 14*x - 47
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assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2*x**2 - 5
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assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10*x**4 + 49
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assert minimal_polynomial(2*I + sqrt(2 + I), x) == x**4 + 4*x**2 + 8*x + 37
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assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10*x**2 + 1
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assert minimal_polynomial(
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sqrt(2) + sqrt(3) + sqrt(6), x) == x**4 - 22*x**2 - 48*x - 23
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a = 1 - 9*sqrt(2) + 7*sqrt(3)
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assert minimal_polynomial(
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1/a, x) == 392*x**4 - 1232*x**3 + 612*x**2 + 4*x - 1
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assert minimal_polynomial(
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1/sqrt(a), x) == 392*x**8 - 1232*x**6 + 612*x**4 + 4*x**2 - 1
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raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
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raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
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raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
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assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
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assert minimal_polynomial(sqrt(2), x) == x**2 - 2
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assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
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assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2, domain='QQ')
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assert minimal_polynomial(sqrt(2), x, polys=True, compose=False) == Poly(x**2 - 2, domain='QQ')
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a = AlgebraicNumber(sqrt(2))
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b = AlgebraicNumber(sqrt(3))
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assert minimal_polynomial(a, x) == x**2 - 2
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assert minimal_polynomial(b, x) == x**2 - 3
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assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2, domain='QQ')
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assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3, domain='QQ')
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assert minimal_polynomial(sqrt(a/2 + 17), x) == 2*x**4 - 68*x**2 + 577
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assert minimal_polynomial(sqrt(b/2 + 17), x) == 4*x**4 - 136*x**2 + 1153
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a, b = sqrt(2)/3 + 7, AlgebraicNumber(sqrt(2)/3 + 7)
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f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
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31608*x**2 - 189648*x + 141358
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assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
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assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
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assert minimal_polynomial(
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a**Q(3, 2), x) == 729*x**4 - 506898*x**2 + 84604519
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# issue 5994
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eq = S('''
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-1/(800*sqrt(-1/240 + 1/(18000*(-1/17280000 +
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sqrt(15)*I/28800000)**(1/3)) + 2*(-1/17280000 +
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sqrt(15)*I/28800000)**(1/3)))''')
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assert minimal_polynomial(eq, x) == 8000*x**2 - 1
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ex = (sqrt(5)*sqrt(I)/(5*sqrt(1 + 125*I))
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+ 25*sqrt(5)/(I**Q(5,2)*(1 + 125*I)**Q(3,2))
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+ 3125*sqrt(5)/(I**Q(11,2)*(1 + 125*I)**Q(3,2))
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+ 5*I*sqrt(1 - I/125))
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mp = minimal_polynomial(ex, x)
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assert mp == 25*x**4 + 5000*x**2 + 250016
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ex = 1 + sqrt(2) + sqrt(3)
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mp = minimal_polynomial(ex, x)
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assert mp == x**4 - 4*x**3 - 4*x**2 + 16*x - 8
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ex = 1/(1 + sqrt(2) + sqrt(3))
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mp = minimal_polynomial(ex, x)
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assert mp == 8*x**4 - 16*x**3 + 4*x**2 + 4*x - 1
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p = (expand((1 + sqrt(2) - 2*sqrt(3) + sqrt(7))**3))**Rational(1, 3)
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mp = minimal_polynomial(p, x)
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assert mp == x**8 - 8*x**7 - 56*x**6 + 448*x**5 + 480*x**4 - 5056*x**3 + 1984*x**2 + 7424*x - 3008
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p = expand((1 + sqrt(2) - 2*sqrt(3) + sqrt(7))**3)
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mp = minimal_polynomial(p, x)
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assert mp == x**8 - 512*x**7 - 118208*x**6 + 31131136*x**5 + 647362560*x**4 - 56026611712*x**3 + 116994310144*x**2 + 404854931456*x - 27216576512
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assert minimal_polynomial(S("-sqrt(5)/2 - 1/2 + (-sqrt(5)/2 - 1/2)**2"), x) == x - 1
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a = 1 + sqrt(2)
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assert minimal_polynomial((a*sqrt(2) + a)**3, x) == x**2 - 198*x + 1
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p = 1/(1 + sqrt(2) + sqrt(3))
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assert minimal_polynomial(p, x, compose=False) == 8*x**4 - 16*x**3 + 4*x**2 + 4*x - 1
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p = 2/(1 + sqrt(2) + sqrt(3))
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assert minimal_polynomial(p, x, compose=False) == x**4 - 4*x**3 + 2*x**2 + 4*x - 2
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assert minimal_polynomial(1 + sqrt(2)*I, x, compose=False) == x**2 - 2*x + 3
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assert minimal_polynomial(1/(1 + sqrt(2)) + 1, x, compose=False) == x**2 - 2
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assert minimal_polynomial(sqrt(2)*I + I*(1 + sqrt(2)), x,
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compose=False) == x**4 + 18*x**2 + 49
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# minimal polynomial of I
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assert minimal_polynomial(I, x, domain=QQ.algebraic_field(I)) == x - I
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K = QQ.algebraic_field(I*(sqrt(2) + 1))
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assert minimal_polynomial(I, x, domain=K) == x - I
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assert minimal_polynomial(I, x, domain=QQ) == x**2 + 1
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assert minimal_polynomial(I, x, domain='QQ(y)') == x**2 + 1
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#issue 11553
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assert minimal_polynomial(GoldenRatio, x) == x**2 - x - 1
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assert minimal_polynomial(TribonacciConstant + 3, x) == x**3 - 10*x**2 + 32*x - 34
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assert minimal_polynomial(GoldenRatio, x, domain=QQ.algebraic_field(sqrt(5))) == \
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2*x - sqrt(5) - 1
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assert minimal_polynomial(TribonacciConstant, x, domain=QQ.algebraic_field(cbrt(19 - 3*sqrt(33)))) == \
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48*x - 19*(19 - 3*sqrt(33))**Rational(2, 3) - 3*sqrt(33)*(19 - 3*sqrt(33))**Rational(2, 3) \
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- 16*(19 - 3*sqrt(33))**Rational(1, 3) - 16
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# AlgebraicNumber with an alias.
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# Wester H24
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phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi')
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assert minimal_polynomial(phi, x) == x**2 - x - 1
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def test_minimal_polynomial_issue_19732():
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# https://github.com/sympy/sympy/issues/19732
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expr = (-280898097948878450887044002323982963174671632174995451265117559518123750720061943079105185551006003416773064305074191140286225850817291393988597615/(-488144716373031204149459129212782509078221364279079444636386844223983756114492222145074506571622290776245390771587888364089507840000000*sqrt(238368341569)*sqrt(S(11918417078450)/63568729
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- 24411360*sqrt(238368341569)/63568729) +
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238326799225996604451373809274348704114327860564921529846705817404208077866956345381951726531296652901169111729944612727047670549086208000000*sqrt(S(11918417078450)/63568729
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- 24411360*sqrt(238368341569)/63568729)) -
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180561807339168676696180573852937120123827201075968945871075967679148461189459480842956689723484024031016208588658753107/(-59358007109636562851035004992802812513575019937126272896569856090962677491318275291141463850327474176000000*sqrt(238368341569)*sqrt(S(11918417078450)/63568729
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- 24411360*sqrt(238368341569)/63568729) +
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28980348180319251787320809875930301310576055074938369007463004788921613896002936637780993064387310446267596800000*sqrt(S(11918417078450)/63568729
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- 24411360*sqrt(238368341569)/63568729)))
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poly = (2151288870990266634727173620565483054187142169311153766675688628985237817262915166497766867289157986631135400926544697981091151416655364879773546003475813114962656742744975460025956167152918469472166170500512008351638710934022160294849059721218824490226159355197136265032810944357335461128949781377875451881300105989490353140886315677977149440000000000000000000000*x**4
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- 5773274155644072033773937864114266313663195672820501581692669271302387257492905909558846459600429795784309388968498783843631580008547382703258503404023153694528041873101120067477617592651525155101107144042679962433039557235772239171616433004024998230222455940044709064078962397144550855715640331680262171410099614469231080995436488414164502751395405398078353242072696360734131090111239998110773292915337556205692674790561090109440000000000000*x**2
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+ 211295968822207088328287206509522887719741955693091053353263782924470627623790749534705683380138972642560898936171035770539616881000369889020398551821767092685775598633794696371561234818461806577723412581353857653829324364446419444210520602157621008010129702779407422072249192199762604318993590841636967747488049176548615614290254356975376588506729604345612047361483789518445332415765213187893207704958013682516462853001964919444736320672860140355089)
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assert minimal_polynomial(expr, x) == poly
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def test_minimal_polynomial_hi_prec():
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p = 1/sqrt(1 - 9*sqrt(2) + 7*sqrt(3) + Rational(1, 10)**30)
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mp = minimal_polynomial(p, x)
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# checked with Wolfram Alpha
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assert mp.coeff(x**6) == -1232000000000000000000000000001223999999999999999999999999999987999999999999999999999999999996000000000000000000000000000000
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def test_minimal_polynomial_sq():
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from sympy.core.add import Add
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from sympy.core.function import expand_multinomial
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p = expand_multinomial((1 + 5*sqrt(2) + 2*sqrt(3))**3)
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mp = minimal_polynomial(p**Rational(1, 3), x)
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assert mp == x**4 - 4*x**3 - 118*x**2 + 244*x + 1321
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p = expand_multinomial((1 + sqrt(2) - 2*sqrt(3) + sqrt(7))**3)
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mp = minimal_polynomial(p**Rational(1, 3), x)
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assert mp == x**8 - 8*x**7 - 56*x**6 + 448*x**5 + 480*x**4 - 5056*x**3 + 1984*x**2 + 7424*x - 3008
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p = Add(*[sqrt(i) for i in range(1, 12)])
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mp = minimal_polynomial(p, x)
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assert mp.subs({x: 0}) == -71965773323122507776
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def test_minpoly_compose():
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# issue 6868
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eq = S('''
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-1/(800*sqrt(-1/240 + 1/(18000*(-1/17280000 +
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sqrt(15)*I/28800000)**(1/3)) + 2*(-1/17280000 +
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sqrt(15)*I/28800000)**(1/3)))''')
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mp = minimal_polynomial(eq + 3, x)
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assert mp == 8000*x**2 - 48000*x + 71999
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# issue 5888
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assert minimal_polynomial(exp(I*pi/8), x) == x**8 + 1
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mp = minimal_polynomial(sin(pi/7) + sqrt(2), x)
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assert mp == 4096*x**12 - 63488*x**10 + 351488*x**8 - 826496*x**6 + \
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770912*x**4 - 268432*x**2 + 28561
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mp = minimal_polynomial(cos(pi/7) + sqrt(2), x)
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assert mp == 64*x**6 - 64*x**5 - 432*x**4 + 304*x**3 + 712*x**2 - \
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232*x - 239
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mp = minimal_polynomial(exp(I*pi/7) + sqrt(2), x)
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assert mp == x**12 - 2*x**11 - 9*x**10 + 16*x**9 + 43*x**8 - 70*x**7 - 97*x**6 + 126*x**5 + 211*x**4 - 212*x**3 - 37*x**2 + 142*x + 127
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mp = minimal_polynomial(sin(pi/7) + sqrt(2), x)
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assert mp == 4096*x**12 - 63488*x**10 + 351488*x**8 - 826496*x**6 + \
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770912*x**4 - 268432*x**2 + 28561
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mp = minimal_polynomial(cos(pi/7) + sqrt(2), x)
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assert mp == 64*x**6 - 64*x**5 - 432*x**4 + 304*x**3 + 712*x**2 - \
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232*x - 239
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mp = minimal_polynomial(exp(I*pi/7) + sqrt(2), x)
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assert mp == x**12 - 2*x**11 - 9*x**10 + 16*x**9 + 43*x**8 - 70*x**7 - 97*x**6 + 126*x**5 + 211*x**4 - 212*x**3 - 37*x**2 + 142*x + 127
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mp = minimal_polynomial(exp(I*pi*Rational(2, 7)), x)
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assert mp == x**6 + x**5 + x**4 + x**3 + x**2 + x + 1
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mp = minimal_polynomial(exp(I*pi*Rational(2, 15)), x)
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assert mp == x**8 - x**7 + x**5 - x**4 + x**3 - x + 1
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mp = minimal_polynomial(cos(pi*Rational(2, 7)), x)
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assert mp == 8*x**3 + 4*x**2 - 4*x - 1
|
||
|
|
mp = minimal_polynomial(sin(pi*Rational(2, 7)), x)
|
||
|
|
ex = (5*cos(pi*Rational(2, 7)) - 7)/(9*cos(pi/7) - 5*cos(pi*Rational(3, 7)))
|
||
|
|
mp = minimal_polynomial(ex, x)
|
||
|
|
assert mp == x**3 + 2*x**2 - x - 1
|
||
|
|
assert minimal_polynomial(-1/(2*cos(pi/7)), x) == x**3 + 2*x**2 - x - 1
|
||
|
|
assert minimal_polynomial(sin(pi*Rational(2, 15)), x) == \
|
||
|
|
256*x**8 - 448*x**6 + 224*x**4 - 32*x**2 + 1
|
||
|
|
assert minimal_polynomial(sin(pi*Rational(5, 14)), x) == 8*x**3 - 4*x**2 - 4*x + 1
|
||
|
|
assert minimal_polynomial(cos(pi/15), x) == 16*x**4 + 8*x**3 - 16*x**2 - 8*x + 1
|
||
|
|
|
||
|
|
ex = rootof(x**3 +x*4 + 1, 0)
|
||
|
|
mp = minimal_polynomial(ex, x)
|
||
|
|
assert mp == x**3 + 4*x + 1
|
||
|
|
mp = minimal_polynomial(ex + 1, x)
|
||
|
|
assert mp == x**3 - 3*x**2 + 7*x - 4
|
||
|
|
assert minimal_polynomial(exp(I*pi/3), x) == x**2 - x + 1
|
||
|
|
assert minimal_polynomial(exp(I*pi/4), x) == x**4 + 1
|
||
|
|
assert minimal_polynomial(exp(I*pi/6), x) == x**4 - x**2 + 1
|
||
|
|
assert minimal_polynomial(exp(I*pi/9), x) == x**6 - x**3 + 1
|
||
|
|
assert minimal_polynomial(exp(I*pi/10), x) == x**8 - x**6 + x**4 - x**2 + 1
|
||
|
|
assert minimal_polynomial(sin(pi/9), x) == 64*x**6 - 96*x**4 + 36*x**2 - 3
|
||
|
|
assert minimal_polynomial(sin(pi/11), x) == 1024*x**10 - 2816*x**8 + \
|
||
|
|
2816*x**6 - 1232*x**4 + 220*x**2 - 11
|
||
|
|
assert minimal_polynomial(sin(pi/21), x) == 4096*x**12 - 11264*x**10 + \
|
||
|
|
11264*x**8 - 4992*x**6 + 960*x**4 - 64*x**2 + 1
|
||
|
|
assert minimal_polynomial(cos(pi/9), x) == 8*x**3 - 6*x - 1
|
||
|
|
|
||
|
|
ex = 2**Rational(1, 3)*exp(2*I*pi/3)
|
||
|
|
assert minimal_polynomial(ex, x) == x**3 - 2
|
||
|
|
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(cos(pi*sqrt(2)), x))
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(sin(pi*sqrt(2)), x))
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(exp(1.618*I*pi), x))
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(exp(I*pi*sqrt(2)), x))
|
||
|
|
|
||
|
|
# issue 5934
|
||
|
|
ex = 1/(-36000 - 7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) +
|
||
|
|
24*sqrt(10)*sqrt(-sqrt(5) + 5))**2) + 1
|
||
|
|
raises(ZeroDivisionError, lambda: minimal_polynomial(ex, x))
|
||
|
|
|
||
|
|
ex = sqrt(1 + 2**Rational(1,3)) + sqrt(1 + 2**Rational(1,4)) + sqrt(2)
|
||
|
|
mp = minimal_polynomial(ex, x)
|
||
|
|
assert degree(mp) == 48 and mp.subs({x:0}) == -16630256576
|
||
|
|
|
||
|
|
ex = tan(pi/5, evaluate=False)
|
||
|
|
mp = minimal_polynomial(ex, x)
|
||
|
|
assert mp == x**4 - 10*x**2 + 5
|
||
|
|
assert mp.subs(x, tan(pi/5)).is_zero
|
||
|
|
|
||
|
|
ex = tan(pi/6, evaluate=False)
|
||
|
|
mp = minimal_polynomial(ex, x)
|
||
|
|
assert mp == 3*x**2 - 1
|
||
|
|
assert mp.subs(x, tan(pi/6)).is_zero
|
||
|
|
|
||
|
|
ex = tan(pi/10, evaluate=False)
|
||
|
|
mp = minimal_polynomial(ex, x)
|
||
|
|
assert mp == 5*x**4 - 10*x**2 + 1
|
||
|
|
assert mp.subs(x, tan(pi/10)).is_zero
|
||
|
|
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(tan(pi*sqrt(2)), x))
|
||
|
|
|
||
|
|
|
||
|
|
def test_minpoly_issue_7113():
|
||
|
|
# see discussion in https://github.com/sympy/sympy/pull/2234
|
||
|
|
from sympy.simplify.simplify import nsimplify
|
||
|
|
r = nsimplify(pi, tolerance=0.000000001)
|
||
|
|
mp = minimal_polynomial(r, x)
|
||
|
|
assert mp == 1768292677839237920489538677417507171630859375*x**109 - \
|
||
|
|
2734577732179183863586489182929671773182898498218854181690460140337930774573792597743853652058046464
|
||
|
|
|
||
|
|
|
||
|
|
def test_minpoly_issue_23677():
|
||
|
|
r1 = CRootOf(4000000*x**3 - 239960000*x**2 + 4782399900*x - 31663998001, 0)
|
||
|
|
r2 = CRootOf(4000000*x**3 - 239960000*x**2 + 4782399900*x - 31663998001, 1)
|
||
|
|
num = (7680000000000000000*r1**4*r2**4 - 614323200000000000000*r1**4*r2**3
|
||
|
|
+ 18458112576000000000000*r1**4*r2**2 - 246896663036160000000000*r1**4*r2
|
||
|
|
+ 1240473830323209600000000*r1**4 - 614323200000000000000*r1**3*r2**4
|
||
|
|
- 1476464424954240000000000*r1**3*r2**2 - 99225501687553535904000000*r1**3
|
||
|
|
+ 18458112576000000000000*r1**2*r2**4 - 1476464424954240000000000*r1**2*r2**3
|
||
|
|
- 593391458458356671712000000*r1**2*r2 + 2981354896834339226880720000*r1**2
|
||
|
|
- 246896663036160000000000*r1*r2**4 - 593391458458356671712000000*r1*r2**2
|
||
|
|
- 39878756418031796275267195200*r1 + 1240473830323209600000000*r2**4
|
||
|
|
- 99225501687553535904000000*r2**3 + 2981354896834339226880720000*r2**2 -
|
||
|
|
39878756418031796275267195200*r2 + 200361370275616536577343808012)
|
||
|
|
mp = (x**3 + 59426520028417434406408556687919*x**2 +
|
||
|
|
1161475464966574421163316896737773190861975156439163671112508400*x +
|
||
|
|
7467465541178623874454517208254940823818304424383315270991298807299003671748074773558707779600)
|
||
|
|
assert minimal_polynomial(num, x) == mp
|
||
|
|
|
||
|
|
|
||
|
|
def test_minpoly_issue_7574():
|
||
|
|
ex = -(-1)**Rational(1, 3) + (-1)**Rational(2,3)
|
||
|
|
assert minimal_polynomial(ex, x) == x + 1
|
||
|
|
|
||
|
|
|
||
|
|
def test_choose_factor():
|
||
|
|
# Test that this does not enter an infinite loop:
|
||
|
|
bad_factors = [Poly(x-2, x), Poly(x+2, x)]
|
||
|
|
raises(NotImplementedError, lambda: _choose_factor(bad_factors, x, sqrt(3)))
|
||
|
|
|
||
|
|
|
||
|
|
def test_minpoly_fraction_field():
|
||
|
|
assert minimal_polynomial(1/x, y) == -x*y + 1
|
||
|
|
assert minimal_polynomial(1 / (x + 1), y) == (x + 1)*y - 1
|
||
|
|
|
||
|
|
assert minimal_polynomial(sqrt(x), y) == y**2 - x
|
||
|
|
assert minimal_polynomial(sqrt(x + 1), y) == y**2 - x - 1
|
||
|
|
assert minimal_polynomial(sqrt(x) / x, y) == x*y**2 - 1
|
||
|
|
assert minimal_polynomial(sqrt(2) * sqrt(x), y) == y**2 - 2 * x
|
||
|
|
assert minimal_polynomial(sqrt(2) + sqrt(x), y) == \
|
||
|
|
y**4 + (-2*x - 4)*y**2 + x**2 - 4*x + 4
|
||
|
|
|
||
|
|
assert minimal_polynomial(x**Rational(1,3), y) == y**3 - x
|
||
|
|
assert minimal_polynomial(x**Rational(1,3) + sqrt(x), y) == \
|
||
|
|
y**6 - 3*x*y**4 - 2*x*y**3 + 3*x**2*y**2 - 6*x**2*y - x**3 + x**2
|
||
|
|
|
||
|
|
assert minimal_polynomial(sqrt(x) / z, y) == z**2*y**2 - x
|
||
|
|
assert minimal_polynomial(sqrt(x) / (z + 1), y) == (z**2 + 2*z + 1)*y**2 - x
|
||
|
|
|
||
|
|
assert minimal_polynomial(1/x, y, polys=True) == Poly(-x*y + 1, y, domain='ZZ(x)')
|
||
|
|
assert minimal_polynomial(1 / (x + 1), y, polys=True) == \
|
||
|
|
Poly((x + 1)*y - 1, y, domain='ZZ(x)')
|
||
|
|
assert minimal_polynomial(sqrt(x), y, polys=True) == Poly(y**2 - x, y, domain='ZZ(x)')
|
||
|
|
assert minimal_polynomial(sqrt(x) / z, y, polys=True) == \
|
||
|
|
Poly(z**2*y**2 - x, y, domain='ZZ(x, z)')
|
||
|
|
|
||
|
|
# this is (sqrt(1 + x**3)/x).integrate(x).diff(x) - sqrt(1 + x**3)/x
|
||
|
|
a = sqrt(x)/sqrt(1 + x**(-3)) - sqrt(x**3 + 1)/x + 1/(x**Rational(5, 2)* \
|
||
|
|
(1 + x**(-3))**Rational(3, 2)) + 1/(x**Rational(11, 2)*(1 + x**(-3))**Rational(3, 2))
|
||
|
|
|
||
|
|
assert minimal_polynomial(a, y) == y
|
||
|
|
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(exp(x), y))
|
||
|
|
raises(GeneratorsError, lambda: minimal_polynomial(sqrt(x), x))
|
||
|
|
raises(GeneratorsError, lambda: minimal_polynomial(sqrt(x) - y, x))
|
||
|
|
raises(NotImplementedError, lambda: minimal_polynomial(sqrt(x), y, compose=False))
|
||
|
|
|
||
|
|
@slow
|
||
|
|
def test_minpoly_fraction_field_slow():
|
||
|
|
assert minimal_polynomial(minimal_polynomial(sqrt(x**Rational(1,5) - 1),
|
||
|
|
y).subs(y, sqrt(x**Rational(1,5) - 1)), z) == z
|
||
|
|
|
||
|
|
def test_minpoly_domain():
|
||
|
|
assert minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2))) == \
|
||
|
|
x - sqrt(2)
|
||
|
|
assert minimal_polynomial(sqrt(8), x, domain=QQ.algebraic_field(sqrt(2))) == \
|
||
|
|
x - 2*sqrt(2)
|
||
|
|
assert minimal_polynomial(sqrt(Rational(3,2)), x,
|
||
|
|
domain=QQ.algebraic_field(sqrt(2))) == 2*x**2 - 3
|
||
|
|
|
||
|
|
raises(NotAlgebraic, lambda: minimal_polynomial(y, x, domain=QQ))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_14831():
|
||
|
|
a = -2*sqrt(2)*sqrt(12*sqrt(2) + 17)
|
||
|
|
assert minimal_polynomial(a, x) == x**2 + 16*x - 8
|
||
|
|
e = (-3*sqrt(12*sqrt(2) + 17) + 12*sqrt(2) +
|
||
|
|
17 - 2*sqrt(2)*sqrt(12*sqrt(2) + 17))
|
||
|
|
assert minimal_polynomial(e, x) == x
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_18248():
|
||
|
|
assert nonlinsolve([x*y**3-sqrt(2)/3, x*y**6-4/(9*(sqrt(3)))],x,y) == \
|
||
|
|
FiniteSet((sqrt(3)/2, sqrt(6)/3), (sqrt(3)/2, -sqrt(6)/6 - sqrt(2)*I/2),
|
||
|
|
(sqrt(3)/2, -sqrt(6)/6 + sqrt(2)*I/2))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_13230():
|
||
|
|
c1 = Circle(Point2D(3, sqrt(5)), 5)
|
||
|
|
c2 = Circle(Point2D(4, sqrt(7)), 6)
|
||
|
|
assert intersection(c1, c2) == [Point2D(-1 + (-sqrt(7) + sqrt(5))*(-2*sqrt(7)/29
|
||
|
|
+ 9*sqrt(5)/29 + sqrt(196*sqrt(35) + 1941)/29), -2*sqrt(7)/29 + 9*sqrt(5)/29
|
||
|
|
+ sqrt(196*sqrt(35) + 1941)/29), Point2D(-1 + (-sqrt(7) + sqrt(5))*(-sqrt(196*sqrt(35)
|
||
|
|
+ 1941)/29 - 2*sqrt(7)/29 + 9*sqrt(5)/29), -sqrt(196*sqrt(35) + 1941)/29 - 2*sqrt(7)/29 + 9*sqrt(5)/29)]
|
||
|
|
|
||
|
|
def test_issue_19760():
|
||
|
|
e = 1/(sqrt(1 + sqrt(2)) - sqrt(2)*sqrt(1 + sqrt(2))) + 1
|
||
|
|
mp_expected = x**4 - 4*x**3 + 4*x**2 - 2
|
||
|
|
|
||
|
|
for comp in (True, False):
|
||
|
|
mp = Poly(minimal_polynomial(e, compose=comp))
|
||
|
|
assert mp(x) == mp_expected, "minimal_polynomial(e, compose=%s) = %s; %s expected" % (comp, mp(x), mp_expected)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_20163():
|
||
|
|
assert apart(1/(x**6+1), extension=[sqrt(3), I]) == \
|
||
|
|
(sqrt(3) + I)/(2*x + sqrt(3) + I)/6 + \
|
||
|
|
(sqrt(3) - I)/(2*x + sqrt(3) - I)/6 - \
|
||
|
|
(sqrt(3) - I)/(2*x - sqrt(3) + I)/6 - \
|
||
|
|
(sqrt(3) + I)/(2*x - sqrt(3) - I)/6 + \
|
||
|
|
I/(x + I)/6 - I/(x - I)/6
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_22559():
|
||
|
|
alpha = AlgebraicNumber(sqrt(2))
|
||
|
|
assert minimal_polynomial(alpha**3, x) == x**2 - 8
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_22561():
|
||
|
|
a = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1) / 2, 0, S(-9) / 2, 0], gen=x)
|
||
|
|
assert a.as_expr() == sqrt(2)
|
||
|
|
assert minimal_polynomial(a, x) == x**2 - 2
|
||
|
|
assert minimal_polynomial(a**3, x) == x**2 - 8
|
||
|
|
|
||
|
|
|
||
|
|
def test_separate_sq_not_impl():
|
||
|
|
raises(NotImplementedError, lambda: _separate_sq(x**(S(1)/3) + x))
|
||
|
|
|
||
|
|
|
||
|
|
def test_minpoly_op_algebraic_element_not_impl():
|
||
|
|
raises(NotImplementedError,
|
||
|
|
lambda: _minpoly_op_algebraic_element(Pow, sqrt(2), sqrt(3), x, QQ))
|
||
|
|
|
||
|
|
|
||
|
|
def test_minpoly_groebner():
|
||
|
|
assert _minpoly_groebner(S(2)/3, x, Poly) == 3*x - 2
|
||
|
|
assert _minpoly_groebner(
|
||
|
|
(sqrt(2) + 3)*(sqrt(2) + 1), x, Poly) == x**2 - 10*x - 7
|
||
|
|
assert _minpoly_groebner((sqrt(2) + 3)**(S(1)/3)*(sqrt(2) + 1)**(S(1)/3),
|
||
|
|
x, Poly) == x**6 - 10*x**3 - 7
|
||
|
|
assert _minpoly_groebner((sqrt(2) + 3)**(-S(1)/3)*(sqrt(2) + 1)**(S(1)/3),
|
||
|
|
x, Poly) == 7*x**6 - 2*x**3 - 1
|
||
|
|
raises(NotAlgebraic, lambda: _minpoly_groebner(pi**2, x, Poly))
|