Coverage for local/lib/python2.7/site-packages/sage/tests/french_book/mpoly.py : 0%

## -*- encoding: utf-8 -*- 

""" 

Doctests from French Sage book 

Test file for chapter "Systèmes polynomiaux" ("Polynomial systems") 

Sage example in ./mpoly.tex, line 35 (in svn rev 1261):: 

sage: R = PolynomialRing(QQ, 'x,y,z') 

sage: x,y,z = R.gens() # donne le n-uplet des indéterminées 

Sage example in ./mpoly.tex, line 40:: 

sage: R = PolynomialRing(QQ, 'x', 10) 

Sage example in ./mpoly.tex, line 44:: 

sage: x = R.gens() 

sage: sum(x[i] for i in range(5)) 

x0 + x1 + x2 + x3 + x4 

Sage example in ./mpoly.tex, line 52:: 

sage: def test_poly(ring, deg=3): 

....: monomials = Subsets( 

....: flatten([(x,)*deg for x in (1,) + ring.gens()]), 

....: deg, submultiset=True) 

....: return add(mul(m) for m in monomials) 

Sage example in ./mpoly.tex, line 59:: 

sage: test_poly(QQ['x,y']) 

x^3 + x^2*y + x*y^2 + y^3 + x^2 + x*y + y^2 + x + y + 1 

sage: test_poly(QQ['y,x']) 

y^3 + y^2*x + y*x^2 + x^3 + y^2 + y*x + x^2 + y + x + 1 

sage: test_poly(QQ['x,y']) == test_poly(QQ['y,x']) 

True 

Sage example in ./mpoly.tex, line 74:: 

sage: test_poly(PolynomialRing(QQ, 'x,y', order='deglex')) 

x^3 + x^2*y + x*y^2 + y^3 + x^2 + x*y + y^2 + x + y + 1 

Sage example in ./mpoly.tex, line 138:: 

sage: R.<x,y> = InfinitePolynomialRing(ZZ, order='lex') 

sage: p = mul(x[k] - y[k] for k in range(2)); p 

x_1*x_0 - x_1*y_0 - x_0*y_1 + y_1*y_0 

sage: p + x[100] 

x_100 + x_1*x_0 - x_1*y_0 - x_0*y_1 + y_1*y_0 

Sage example in ./mpoly.tex, line 188:: 

sage: R.<x,y,z> = QQ[] 

sage: p = 7*y^2*x^2 + 3*y*x^2 + 2*y*z + x^3 + 6 

sage: p.lt() 

7*x^2*y^2 

Sage example in ./mpoly.tex, line 196:: 

sage: p[x^2*y] == p[(2,1,0)] == p[2,1,0] == 3 

True 

Sage example in ./mpoly.tex, line 202:: 

sage: p(0, 3, -1) 

0 

sage: p.subs(x = 1, z = x^2+1) 

2*x^2*y + 7*y^2 + 5*y + 7 

Sage example in ./mpoly.tex, line 209:: 

sage: print("total={d} (en x)={dx} partiels={ds}" 

....: .format(d=p.degree(), dx=p.degree(x), ds=p.degrees())) 

total=4 (en x)=3 partiels=(3, 2, 1) 

Sage example in ./mpoly.tex, line 255:: 

sage: R.<x,y> = QQ[]; p = x^2 + y^2; q = x + y 

sage: print("({quo})*({q}) + ({rem}) == {p}".format( \ 

....: quo=p//q, q=q, rem=p%q, p=p//q*q+p%q)) 

(-x + y)*(x + y) + (2*x^2) == x^2 + y^2 

sage: p.mod(q) # n'est PAS équivalent à p%q 

2*y^2 

Sage example in ./mpoly.tex, line 277:: 

sage: k.<a> = GF(9); R.<x,y,z> = k[] 

sage: (a*x^2*z^2 + x*y*z - y^2).factor(proof=False) 

((a)) * (x*z + (-a - 1)*y) * (x*z + (-a)*y) 

Sage example in ./mpoly.tex, line 325:: 

sage: R.<x,y,z> = QQ[] 

sage: J = R.ideal(x^2 * y * z - 18, 

....: x * y^3 * z - 24, 

....: x * y * z^4 - 6); 

Sage example in ./mpoly.tex, line 333:: 

sage: J.dimension() 

0 

Sage example in ./mpoly.tex, line 339:: 

sage: J.variety() 

[{y: 2, z: 1, x: 3}] 

Sage example in ./mpoly.tex, line 347:: 

sage: V = J.variety(QQbar) 

sage: len(V) 

17 

Sage example in ./mpoly.tex, line 353:: 

sage: V[-3:] 

[{z: 0.9324722294043558? - 0.3612416661871530?*I, 

y: -1.700434271459229? + 1.052864325754712?*I, 

x: 1.337215067329615? - 2.685489874065187?*I}, 

{z: 0.9324722294043558? + 0.3612416661871530?*I, 

y: -1.700434271459229? - 1.052864325754712?*I, 

x: 1.337215067329615? + 2.685489874065187?*I}, 

{z: 1, y: 2, x: 3}] 

Sage example in ./mpoly.tex, line 364:: 

sage: (xx, yy, zz) = QQbar['x,y,z'].gens() 

sage: [ pt[xx].degree() for pt in V ] 

[16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 

16, 1] 

Sage example in ./mpoly.tex, line 376:: 

sage: Set(tuple(abs(pt[i]) for i in (xx,yy,zz)) for pt in V) 

{(3, 2, 1)} 

Sage example in ./mpoly.tex, line 387:: 

sage: w = QQbar.zeta(17); w # racine primitive de 1 

0.9324722294043558? + 0.3612416661871530?*I 

sage: Set(pt[zz] for pt in V) == Set(w^i for i in range(17)) 

True 

Sage example in ./mpoly.tex, line 401:: 

sage: set(pt[zz].minpoly() for pt in V[:-1]) 

{x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1} 

Sage example in ./mpoly.tex, line 413:: 

sage: def polar_form(z): 

....: rho = z.abs(); rho.simplify() 

....: theta = 2 * pi * z.rational_argument() 

....: return (SR(rho) * exp(I*theta)) 

sage: [tuple(polar_form(pt[i]) for i in [xx,yy,zz]) 

....: for pt in V[-3:]] 

[(3*e^(-6/17*I*pi), 2*e^(14/17*I*pi), e^(-2/17*I*pi)), 

(3*e^(6/17*I*pi), 2*e^(-14/17*I*pi), e^(2/17*I*pi)), (3, 2, 1)] 

Sage example in ./mpoly.tex, line 432:: 

sage: J.triangular_decomposition() 

[Ideal (z^17 - 1, y - 2*z^10, x - 3*z^3) of Multivariate 

Polynomial Ring in x, y, z over Rational Field] 

sage: J.transformed_basis() 

[z^17 - 1, -2*z^10 + y, -3/4*y^2 + x] 

Sage example in ./mpoly.tex, line 534:: 

sage: R.<x,y> = QQ[] 

sage: J = R.ideal(x^2 + y^2 - 1, 16*x^2*y^2 - 1) 

Sage example in ./mpoly.tex, line 539:: 

sage: ybar2 = R.quo(J)(y^2) 

sage: [ybar2^i for i in range(3)] 

[1, ybar^2, ybar^2 - 1/16] 

sage: ((ybar2 + 1)^2).lift() 

3*y^2 + 15/16 

Sage example in ./mpoly.tex, line 561:: 

sage: u = (16*y^4 - 16*y^2 + 1).lift(J); u 

[16*y^2, -1] 

sage: u[0]*J.0 + u[1]*J.1 

16*y^4 - 16*y^2 + 1 

Sage example in ./mpoly.tex, line 569:: 

sage: (y^4).mod(J) 

y^2 - 1/16 

Sage example in ./mpoly.tex, line 575:: 

sage: (y^4).reduce([x^2 + y^2 - 1, 16*x^2*y^2 - 1]) 

y^4 

Sage example in ./mpoly.tex, line 629:: 

sage: 1 in ideal(x^2+y^2-1, (x-4)^2+y^2-1) 

False 

Sage example in ./mpoly.tex, line 634:: 

sage: R(1).lift(ideal(x^2+y^2-1, (x-4)^2+y^2-1, x-y)) 

[-1/28*y + 1/14, 1/28*y + 1/14, -1/7*x + 1/7*y + 4/7] 

Sage example in ./mpoly.tex, line 650:: 

sage: J1 = (x^2 + y^2 - 1, 16*x^2*y^2 - 1)*R 

sage: J2 = (x^2 + y^2 - 1, 4*x^2*y^2 - 1)*R 

sage: J1.radical() == J1 

True 

sage: J2.radical() 

Ideal (2*y^2 - 1, 2*x^2 - 1) of Multivariate Polynomial 

Ring in x, y over Rational Field 

sage: 2*y^2 - 1 in J2 

False 

Sage example in ./mpoly.tex, line 680:: 

sage: C = ideal(x^2 + y^2 - 1); H = ideal(16*x^2*y^2 - 1) 

sage: C + H == J1 

True 

Sage example in ./mpoly.tex, line 697:: 

sage: CH = C.intersection(H).quotient(ideal(4*x*y-1)); CH 

Ideal (4*x^3*y + 4*x*y^3 + x^2 - 4*x*y + y^2 - 1) of 

Multivariate Polynomial Ring in x, y over Rational Field 

sage: CH.gen(0).factor() 

(4*x*y + 1) * (x^2 + y^2 - 1) 

Sage example in ./mpoly.tex, line 705:: 

sage: H.quotient(C) == H 

True 

Sage example in ./mpoly.tex, line 720:: 

sage: [J.dimension() for J in [J1, J2, C, H, H*J2, J1+J2]] 

[0, 0, 1, 1, 1, -1] 

Sage example in ./mpoly.tex, line 780:: 

sage: R.<x,y,z> = QQ[] 

sage: J = ideal(2*x+y-2*z, 2*x+2*y+z-1) 

sage: J.elimination_ideal(x) 

Ideal (y + 3*z - 1) of Multivariate Polynomial Ring in x, y, z 

over Rational Field 

sage: J.elimination_ideal([x,y]).gens() 

[0] 

Sage example in ./mpoly.tex, line 794:: 

sage: J1.gens() 

[x^2 + y^2 - 1, 16*x^2*y^2 - 1] 

Sage example in ./mpoly.tex, line 858:: 

sage: R.<x,y,t> = QQ[] 

sage: Param = R.ideal((1-t^2)-(1+t^2)*x, 2*t-(1+t^2)*y) 

Sage example in ./mpoly.tex, line 863:: 

sage: Param.elimination_ideal(t).gens() 

[x^2 + y^2 - 1] 

Sage example in ./mpoly.tex, line 886:: 

sage: R.<x,y,t> = QQ[] 

sage: eq = x^2 + (y-t)^2 - 1/2*(t^2+1) 

sage: fig = add((eq(t=k/5)*QQ[x,y]).plot() for k in (-15..15)) 

sage: fig.show(aspect_ratio=1,xmin=-2,xmax=2,ymin=-3,ymax=3) 

Sage example in ./mpoly.tex, line 900:: 

sage: env = ideal(eq, eq.derivative(t)).elimination_ideal(t) 

sage: env.gens() 

[2*x^2 - 2*y^2 - 1] 

Sage example in ./mpoly.tex, line 906:: 

sage: env.change_ring(QQ[x,y]).plot() 

Graphics object consisting of 1 graphics primitive 

Sage example in ./mpoly.tex, line 933:: 

sage: R.<x,y,t> = QQ[] 

sage: J = (y-t*x, y-t*(1-x))*R 

sage: (x^2+y^2) - ((1-x)^2+y^2) in J 

False 

Sage example in ./mpoly.tex, line 942:: 

sage: R.<x,y,t,u> = QQ[] 

sage: J = (y-t*x, y-t*(1-x), t*u-1)*R 

sage: (x^2+y^2) - ((1-x)^2+y^2) in J 

True 

Sage example in ./mpoly.tex, line 965:: 

sage: R.<x,y,t> = QQ[] 

Sage example in ./mpoly.tex, line 968:: 

sage: eq.derivative(t).resultant(eq, t) 

x^2 - y^2 - 1/2 

Sage example in ./mpoly.tex, line 982:: 

sage: R.<x,y> = QQ[] 

sage: ((x^2 + y^2)*(x^2 + y^2 + 1)*R).dimension() 

1 

Sage example in ./mpoly.tex, line 996:: 

sage: J1.variety() 

[] 

Sage example in ./mpoly.tex, line 1002:: 

sage: J1.variety(QQbar)[0:2] 

[{y: -0.9659258262890683?, x: -0.2588190451025208?}, 

{y: -0.9659258262890683?, x: 0.2588190451025208?}] 

Sage example in ./mpoly.tex, line 1037:: 

sage: R.<x,y> = PolynomialRing(QQ, order='lex') 

sage: C = ideal(x^2+y^2-1) 

sage: D = ideal((x+y-1)*(x+y+1)) 

sage: J = C + D 

Sage example in ./mpoly.tex, line 1054:: 

sage: J.triangular_decomposition() 

[Ideal (y, x^2 - 1) of Multivariate Polynomial Ring in x, y 

over Rational Field, 

Ideal (y^2 - 1, x) of Multivariate Polynomial Ring in x, y 

over Rational Field] 

Sage example in ./mpoly.tex, line 1094:: 

sage: D = ideal((x+2*y-1)*(x+2*y+1)); J = C + D 

sage: J.variety() 

[{y: -4/5, x: 3/5}, {y: 0, x: -1}, {y: 0, x: 1}, {y: 4/5, x: -3/5}] 

sage: [ T.gens() for T in J.triangular_decomposition()] 

[[y, x^2 - 1], [25*y^2 - 16, 4*x + 3*y]] 

Sage example in ./mpoly.tex, line 1104:: 

sage: Jy = J.elimination_ideal(x); Jy.gens() 

[25*y^3 - 16*y] 

Sage example in ./mpoly.tex, line 1109:: 

sage: ys = QQ['y'](Jy.0).roots(); ys 

[(4/5, 1), (0, 1), (-4/5, 1)] 

sage: QQ['x'](J.1(y=ys[0][0])).roots() 

[(-3/5, 1), (-13/5, 1)] 

Sage example in ./mpoly.tex, line 1119 (edited manually):: 

sage: ys = CDF['y'](Jy.0).roots(); ys # abs tol 3e-16 

[(-0.8000000000000002, 1), (0.0, 1), (0.8, 1)] 

sage: [CDF['x'](p(y=ys[0][0])).roots() for p in J.gens()] # abs tol 1e-14 

[[(-0.5999999999999999 - 1.306289919090511e-16*I, 1), (0.6000000000000001 + 1.3062899190905113e-16*I, 1)], [(0.6000000000000001 - 3.135095805817224e-16*I, 1), (2.600000000000001 + 3.1350958058172237e-16*I, 1)]] 

Sage example in ./mpoly.tex, line 1135:: 

sage: R.<x,y> = QQ[]; J = ideal([ x^7-(100*x-1)^2, y-x^7+1 ]) 

Sage example in ./mpoly.tex, line 1138:: 

sage: J.variety(AA) 

[{x: 0.00999999900000035?, y: -0.999999999999990?}, 

{x: 0.01000000100000035?, y: -0.999999999999990?}, 

{x: 6.305568998641385?, y: 396340.8901665450?}] 

Sage example in ./mpoly.tex, line 1170:: 

sage: len(J2.variety(QQbar)), J2.vector_space_dimension() 

(4, 8) 

Sage example in ./mpoly.tex, line 1177:: 

sage: J2.normal_basis() 

[x*y^3, y^3, x*y^2, y^2, x*y, y, x, 1] 

Sage example in ./mpoly.tex, line 1325:: 

sage: R.<x,y,z,t> = PolynomialRing(QQ, order='lex') 

Sage example in ./mpoly.tex, line 1343:: 

sage: ((x+y+z)^2).reduce([x-t, y-t^2, z^2-t]) 

2*z*t^2 + 2*z*t + t^4 + 2*t^3 + t^2 + t 

Sage example in ./mpoly.tex, line 1364:: 

sage: R.<x,y> = PolynomialRing(QQ, order='lex') 

sage: (g, h) = (x-y, x-y^2); p = x*y - x 

sage: p.reduce([g, h]) # deux réductions par h 

y^3 - y^2 

sage: p.reduce([h, g]) # deux réductions par g 

y^2 - y 

Sage example in ./mpoly.tex, line 1373:: 

sage: p - y*g + h 

0 

Sage example in ./mpoly.tex, line 1572:: 

sage: R.<x,y> = PolynomialRing(QQ, order='lex') 

sage: R.ideal(x*y^4, x^2*y^3, x^4*y, x^5).basis_is_groebner() 

True 

Sage example in ./mpoly.tex, line 1578:: 

sage: R.ideal(x^2+y^2-1, 16*x^2*y^2-1).basis_is_groebner() 

False 

Sage example in ./mpoly.tex, line 1593:: 

sage: R.ideal(x^2+y^2-1, 16*x^2*y^2-1).groebner_basis() 

[x^2 + y^2 - 1, y^4 - y^2 + 1/16] 

Sage example in ./mpoly.tex, line 1598:: 

sage: R.ideal(16*x^2*y^2-1).groebner_basis() 

[x^2*y^2 - 1/16] 

Sage example in ./mpoly.tex, line 1603:: 

sage: R.ideal(x^2+y^2-1, (x+y)^2-1).groebner_basis() 

[x^2 + y^2 - 1, x*y, y^3 - y] 

Sage example in ./mpoly.tex, line 1609:: 

sage: R_lex.<x,y> = PolynomialRing(QQ, order='lex') 

sage: J_lex = (x*y+x+y^2+1,x^2*y+x*y^2+1)*R_lex; J_lex.gens() 

[x*y + x + y^2 + 1, x^2*y + x*y^2 + 1] 

sage: J_lex.groebner_basis() 

[x - 1/2*y^3 + y^2 + 3/2, y^4 - y^3 - 3*y - 1] 

Sage example in ./mpoly.tex, line 1616:: 

sage: R_invlex = PolynomialRing(QQ, 'x,y', order='invlex') 

sage: J_invlex = J_lex.change_ring(R_invlex); J_invlex.gens() 

[y^2 + x*y + x + 1, x*y^2 + x^2*y + 1] 

sage: J_invlex.groebner_basis() 

[y^2 + x*y + x + 1, x^2 + x - 1] 

Sage example in ./mpoly.tex, line 1623:: 

sage: R_drl = PolynomialRing(QQ, 'x,y', order='degrevlex') 

sage: J_drl = J_lex.change_ring(R_drl); J_drl.gens() 

[x*y + y^2 + x + 1, x^2*y + x*y^2 + 1] 

sage: J_drl.groebner_basis() 

[y^3 - 2*y^2 - 2*x - 3, x^2 + x - 1, x*y + y^2 + x + 1] 

Sage example in ./mpoly.tex, line 1662:: 

sage: p = (x + y)^5 

sage: J_lex.reduce(p) 

17/2*y^3 - 12*y^2 + 4*y - 49/2 

Sage example in ./mpoly.tex, line 1668:: 

sage: p.reduce(J_lex.groebner_basis()) 

17/2*y^3 - 12*y^2 + 4*y - 49/2 

Sage example in ./mpoly.tex, line 1673:: 

sage: R_lex.quo(J_lex)(p) 

17/2*ybar^3 - 12*ybar^2 + 4*ybar - 49/2 

Sage example in ./mpoly.tex, line 1678:: 

sage: R_drl.quo(J_drl)(p) 

5*ybar^2 + 17*xbar + 4*ybar + 1 

Sage example in ./mpoly.tex, line 1685:: 

sage: J_lex.normal_basis() 

[y^3, y^2, y, 1] 

sage: J_invlex.normal_basis() 

[x*y, y, x, 1] 

sage: J_drl.normal_basis() 

[y^2, y, x, 1] 

Sage example in ./mpoly.tex, line 1701:: 

sage: ideal(16*x^2*y^2-1).dimension() 

1 

Sage example in ./mpoly.tex, line 1736:: 

sage: R.<t,x,y,z> = PolynomialRing(QQ, order='lex') 

sage: J = R.ideal( t+x+y+z-1, t^2-x^2-y^2-z^2-1, t-x*y) 

sage: [u.polynomial(u.variable(0)) for u in J.groebner_basis()] 

[t + x + y + z - 1, 

(y + 1)*x + y + z - 1, 

(z - 2)*x + y*z - 2*y - 2*z + 1, 

(z - 2)*y^2 + (-2*z + 1)*y - z^2 + z - 1] 

Sage example in ./mpoly.tex, line 1801:: 

sage: from sage.rings.ideal import Cyclic 

sage: Cyclic(QQ['x,y,z']) 

Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of 

Multivariate Polynomial Ring in x, y, z over Rational Field 

Sage example in ./mpoly.tex, line 1808:: 

sage: def C(R, n): return Cyclic(PolynomialRing(R, 'x', n)) 

Sage example in ./mpoly.tex, line 1822:: 

sage: p = previous_prime(2^30) 

sage: len(C(GF(p), 6).groebner_basis()) 

45 

""" 

from sage.all_cmdline import * # import sage library