Python PolyRing.gens方法代码示例

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
 def testRingQQ(self):
     r = PolyRing( QQ(), "(t,x)", Order.INVLEX );
     self.assertEqual(str(r),'PolyRing(QQ(),"t,x",Order.INVLEX)');
     [one,x,t] = r.gens();
     self.assertTrue(one.isONE());
     self.assertTrue(len(x)==1);
     self.assertTrue(len(t)==1);
     #
     f = 11 * x**4 - 13 * t * x**2 - 11 * x**2 + 2 * t**2 + 11 * t;
     f = f**2 + f + 3;
     #print "f = " + str(f);
     self.assertEqual(str(f),'( 4 * x**4 - 52 * t**2 * x**3 + 44 * x**3 + 213 * t**4 * x**2 - 330 * t**2 * x**2 + 123 * x**2 - 286 * t**6 * x + 528 * t**4 * x - 255 * t**2 * x + 11 * x + 121 * t**8 - 242 * t**6 + 132 * t**4 - 11 * t**2 + 3 )');

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
 def testRingZM(self):
     r = PolyRing( GF(17), "(t,x)", Order.INVLEX );
     self.assertEqual(str(r),'PolyRing(GFI(17),"t,x",Order.INVLEX)');
     [one,x,t] = r.gens();
     self.assertTrue(one.isONE());
     self.assertTrue(len(x)==1);
     self.assertTrue(len(t)==1);
     #
     f = 11 * x**4 - 13 * t * x**2 - 11 * x**2 + 2 * t**2 + 11 * t;
     f = f**2 + f + 3;
     #print "f = " + str(f);
     self.assertEqual(str(f),'( 4 * x**4 + 16 * t**2 * x**3 + 10 * x**3 + 9 * t**4 * x**2 + 10 * t**2 * x**2 + 4 * x**2 + 3 * t**6 * x + t**4 * x + 11 * x + 2 * t**8 + 13 * t**6 + 13 * t**4 + 6 * t**2 + 3 )');

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
#

import sys;

from jas import Ring, PolyRing, ParamIdeal, QQ
from jas import startLog, terminate


# Raksanyi & Walter example
# integral/rational function coefficients


r = PolyRing(PolyRing(QQ(),"a1,a2,a3,a4",PolyRing.grad),"x1,x2,x3,x4",PolyRing.lex);
#print "r  = " + str(r);

[one,a1,a2,a3,a4,x1,x2,x3,x4] = r.gens();

pl = [ ( x4 - ( a4 - a2 ) ),
      ( x1 + x2 + x3 + x4 - ( a1 + a3 + a4 ) ),
      ( x1 * x3 + x1 * x4 + x2 * x3 + x3 * x4 - ( a1 * a4 + a1 * a3 + a3 * a4 ) ),
      ( x1 * x3 * x4 - ( a1 * a3 * a4 ) ) 
     ];
f = ParamIdeal(r,list=pl);
print "ParamIdeal: " + str(f);

gs = f.CGBsystem();
#print "CGBsystem: " + str(gs);
#print;

print f.CGB();

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
from java.lang import System
from java.lang import Integer

from jas import Ring
from jas import PolyRing
from jas import Ideal
from jas import ZM, QQ, AN, RF
from jas import terminate
from jas import startLog

# polynomial examples: factorization over Z_p(x)(sqrt3(x))[y]

Q = PolyRing(ZM(5),"x",PolyRing.lex);
print "Q     = " + str(Q);
[e,a] = Q.gens();
#print "e     = " + str(e);
print "a     = " + str(a);

Qr = RF(Q);
print "Qr    = " + str(Qr.factory());
[er,ar] = Qr.gens();
#print "er    = " + str(er);
#print "ar    = " + str(ar);
print;

Qwx = PolyRing(Qr,"wx",PolyRing.lex);
print "Qwx   = " + str(Qwx);
[ewx,ax,wx] = Qwx.gens();
#print "ewx   = " + str(ewx);
print "ax    = " + str(ax);

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
#

import sys

from java.lang import System

from jas import PolyRing, Ideal
from jas import QQ, AN, RF
from jas import terminate, startLog

# polynomial examples: prime/primary decomposition in Q[w2,x,wx,y,z]

Q = PolyRing(QQ(), "w2,x,wx,y,z", PolyRing.lex)
print "Q     = " + str(Q)

[e, w2, x, wx, y, z] = Q.gens()
print "e     = " + str(e)
print "w2    = " + str(w2)
print "x     = " + str(x)
print "wx    = " + str(wx)
print "y     = " + str(y)
print "z     = " + str(z)
print

w1 = w2 ** 2 - 2
w2 = wx ** 2 - x
f1 = (y ** 2 - x) * (y ** 2 - 2)
# f1 = ( y**2 - x )**3 * ( y**2 - 2 )**2;
f2 = z ** 2 - y ** 2

print "w1 = ", w1

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
print "o1 = " + str(o1);
o2 = (1/o1)**2;
print "o2 = " + str(o2);
o3 = o2 * o1 * o1;
print "o3 = " + str(o3);
o4 = (-69,20164)*oneOR + (-3,20164)*IOR + (-1,5041)*JOR + (-5,20164)*KOR  + (-3,10082)*oneOI + (-7,20164)*IOI + (-2,5041)*JOI + (-9,20164)*KOI;
print "o4 = " + str(o4);
o5 = o2 - o4;
print "o5  = " + str(o5);
print;


print "------- PolyRing(ZZ(),\"x,y,z\") ---------";
r = PolyRing(ZZ(),"x,y,z",PolyRing.grad);
print "r = " + str(r);
[one,x,y,z] = r.gens();
print "one = " + str(one);
print "x   = " + str(x);
print "y   = " + str(y);
print "z   = " + str(z);
p1 = 2 + 3 * x + 4 * y + 5 * z + ( x + y + z )**2;
print "p1  = " + str(p1);
p2  = z**2 + 2 * y * z + 2 * x * z + y**2 + 2 * x * y + x**2 + 5 * z + 4 * y + 3 * x + 2;
print "p2  = " + str(p2);
p3 = p1 - p2;
print "p3  = " + str(p3);
print "p3.factory() = " + str(p3.factory());
print;


print "------- PolyRing(QQ(),\"x,y,z\") ---------";

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
print "zrelations: = " + str( [ str(r) for r in zrelations ] );
print;

pz = SolvPolyRing(QQ(), "x,y,z,t", PolyRing.lex, zrelations);
print "SolvPolyRing: " + str(pz);
print;

pzq = SRF(pz);
print "SolvableQuotientRing: " + str(pzq.ring.toScript); # + ", assoz: " + str(pzq::ring.isAssociative);
#print "gens =" + str( [ str(r) for r in pzq.gens() ] );
print;

pct = PolyRing(pzq,"u,v,w", PolyRing.lex);
#is automatic: [one,x,y,z,t,u,v,w] = p.gens();
print "tgens = " + str( [ str(r) for r in pct.gens() ] );
print;

relations = [#w, v,  v * w - u,
             v, u,  v * u + x,
             w, y,  y * w + y,
             w, z,  z * w - z
            ];

print "relations: = " + str( [ str(r) for r in relations ] );
print;

#startLog();

pt = SolvPolyRing(pzq, "u,v,w", PolyRing.lex, relations);
print "SolvPolyRing: " + str(pt); # + ", is assoz: " + str(pt.ring.isAssociative);

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
from jas import Ring, PolyRing
from jas import terminate, startLog, noThreads

from jas import QQ, ZM, RF, AN

# polynomial examples: ideal radical decomposition, modified from example 8.16 in GB book

# noThreads(); # must be called very early

prime = 5;
cf = ZM(prime);
#cf = QQ();

ca = PolyRing(cf,"a",PolyRing.lex);
#print "ca = " + str(ca);
[ea,aa] = ca.gens();
print "ea   = " + str(ea);
print "aa   = " + str(aa);
print;

#!#roota = aa**prime + 2;
roota = aa**2 + 2;
print "roota = " + str(roota);
Q3a = AN(roota,field=True);
print "Q3a   = " + str(Q3a.factory());

## Q3a = RF(ca);
#print Q3a.gens();

[ea2,aa2] = Q3a.gens();
print "ea2  = " + str(ea2);

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
from java.lang import System

from jas import Ring, PolyRing, QQ, ZM, RF, AN, GF
from jas import terminate, startLog, noThreads

# polynomial examples: ideal radical decomposition, example 8.16 in GB book, base field with p-th root

# noThreads(); # must be called very early

prime = 5;
cf = GF(prime);
#cf = QQ();

ca = PolyRing(cf,"t",PolyRing.lex);
print "ca = " + str(ca);
[ea,ta] = ca.gens();
print "ea   = " + str(ea);
print "ta   = " + str(ta);
print;

Qpt = RF(ca);
#print Qpt.gens();

[ea2,ta2] = Qpt.gens();
print "ea2  = " + str(ea2);
print "ta2  = " + str(ta2);
print;

cr = PolyRing(Qpt,"wpt",PolyRing.lex);
print "polynomial quotient ring: " + str(cr);

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
from basic_sigbased_gb import ggv, ggv_first_implementation
from basic_sigbased_gb import coeff_free_sigbased_gb
from basic_sigbased_gb import arris_algorithm, min_size_mons
from basic_sigbased_gb import f5, f5z

from staggered_linear_basis import staglinbasis


#r = PolyRing( QQ(), "(B,S,T,Z,P,W)", PolyRing.lex );
#r = PolyRing( ZZ(), "(B,S,T,Z,P,W)", PolyRing.lex );
r = PolyRing( ZM(32003), "(B,S,T,Z,P,W)", PolyRing.lex );
#r = PolyRing( ZM(19), "(B,S,T,Z,P,W)", PolyRing.lex );
print "Ring: " + str(r);
print;

[one,B,S,T,Z,P,W] = r.gens();

p1 = 45 * P + 35 * S - 165 * B - 36; 
p2 = 35 * P + 40 * Z + 25 * T - 27 * S;
p3 = 15 * W + 25 * S * P + 30 * Z - 18 * T - 165 * B**2; 
p4 = -9 * W + 15 * T * P + 20 * S * Z; 
p5 = P * W + 2 * T * Z - 11 * B**3;
p6 = 99 * W - 11 * B * S + 3 * B**2;
p7 = 10000 * B**2 + 6600 * B + 2673;

F = [p1,p2,p3,p4,p5,p6,p7];
#F = [p1,p2,p3,p4,p5,p6];

f = r.ideal( list=F );
print "Ideal: " + str(f);
print;

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
from java.lang import System
from java.lang import Integer

from jas import Ring
from jas import PolyRing
from jas import Ideal
from jas import ZM, QQ, AN, RF
from jas import terminate
from jas import startLog

# polynomial examples: factorization over Z_p(sqrt(2))(x)(sqrt(x))[y]

Q = PolyRing(ZM(5), "w2", PolyRing.lex)
print "Q     = " + str(Q)
[e, a] = Q.gens()
# print "e     = " + str(e);
print "a     = " + str(a)

root = a ** 2 - 2
print "root  = " + str(root)
Q2 = AN(root, field=True)
print "Q2    = " + str(Q2.factory())
[one, w2] = Q2.gens()
# print "one   = " + str(one);
# print "w2    = " + str(w2);
print

Qp = PolyRing(Q2, "x", PolyRing.lex)
print "Qp    = " + str(Qp)
[ep, wp, ap] = Qp.gens()

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
#print "Ring: " + str(r);
#print;

#r = PolyRing(ZZ(),"B,S,T,Z,P,W",PolyRing.lex);
#r = PolyRing(QQ(),"B,S,T,Z,P,W",PolyRing.lex);
#r = PolyRing(CC(),"B,S,T,Z,P,W",PolyRing.lex);
#r = PolyRing(DD(),"B,S,T,Z,P,W",PolyRing.lex);
#r = PolyRing(ZM(19),"B,S,T,Z,P,W",PolyRing.lex);
#r = PolyRing(ZM(1152921504606846883),"B,S,T,Z,P,W",PolyRing.lex); # 2^60-93
#rc = PolyRing(ZZ(),"e,f",PolyRing.lex);
#rc = PolyRing(QQ(),"e,f",PolyRing.lex);
#r = PolyRing(rc,"B,S,T,Z,P,W",PolyRing.lex);

rqc = PolyRing(ZZ(),"e,f",PolyRing.lex);
print "Q-Ring: " + str(rqc);
print "rqc.gens() = ", [ str(f) for f in rqc.gens() ];
print;
[pone,pe,pf] = rqc.gens();

r = PolyRing(RF(rqc),"B,S,T,Z,P,W",PolyRing.lex);
print "Ring: " + str(r);
print;

# sage like: with generators for the polynomial ring
print "r.gens() = ", [ str(f) for f in r.gens() ];
print;
[one,e,f,B,S,T,Z,P,W] = r.gens();
#[one,B,S,T,Z,P,W] = r.gens();
#[one,I,B,S,T,Z,P,W] = r.gens();

f1 = 45 * P + 35 * S - 165 * B - 36;

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
from jas import Ring, PolyRing
from jas import terminate
from jas import startLog

from jas import QQ, DD

# polynomial examples: real roots over Q for zero dimensional ideal `cyclic5'

# r = Ring( "Q(x) L" );
r = PolyRing(QQ(), "a,b,c,d,e", PolyRing.lex)

print "Ring: " + str(r)
print

[one, a, b, c, d, e] = r.gens()

f1 = a + b + c + d + e
f2 = a * b + b * c + c * d + d * e + e * a
f3 = a * b * c + b * c * d + c * d * e + d * e * a + e * a * b
f4 = a * b * c * d + b * c * d * e + c * d * e * a + d * e * a * b + e * a * b * c
f5 = a * b * c * d * e - 1

print "f1 = ", f1
print "f2 = ", f2
print "f3 = ", f3
print "f4 = ", f4
print "f5 = ", f5
print

F = r.ideal(list=[f1, f2, f3, f4, f5])

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
# $Id$
#

import sys

from java.lang import System

from jas import Ring, PolyRing
from jas import QQ, AN
from jas import terminate, startLog

# polynomial examples: absolute factorization over Q(i)

Qr = PolyRing(QQ(), "i", PolyRing.lex)
print "Qr    = " + str(Qr)
[e, a] = Qr.gens()
print "e     = " + str(e)
print "a     = " + str(a)
print

imag = a ** 2 + 1
print "imag  = " + str(imag)
Qi = AN(imag, field=True)
print "Qi    = " + str(Qi.factory())
[one, i] = Qi.gens()
print "one   = " + str(one)
print "i     = " + str(i)
print

r = PolyRing(Qi, "x", PolyRing.lex)
print "r    = " + str(r)

# 需要导入模块: from jas import PolyRing [as 别名]
# 或者: from jas.PolyRing import gens [as 别名]
import sys;

from jas import PolyRing, QQ, RF
from jas import startLog
from jas import terminate


# Montes JSC 2002, 33, 183-208, example 11.1
# integral function coefficients

r = PolyRing( PolyRing(QQ(),"c, b, a",PolyRing.lex), "z,y,x", PolyRing.lex );
print "Ring: " + str(r);
print;

#automatic: [one,c,b,a,z,y,x] = r.gens();
print "gens: ", [ str(f) for f in r.gens() ];
print;

f1 =     x + c * y + b * z + a;
f2 = c * x +     y + a * z + b;
f3 = b * x + a * y +     z + c;

F = [f1,f2,f3];

print "F: ", [ str(f) for f in F ];
print;

#startLog();

If = r.paramideal( "", list = F );
print "ParamIdeal: " + str(If);