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 )');
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 )');
import sys; from jas import PolyRing, QQ, RF from jas import Ideal from jas import startLog from jas import terminate # Montes JSC 2002, 33, 183-208, example 11.4 # integral function coefficients R = PolyRing( PolyRing(QQ(),"Q2, P2, Q1, P1",PolyRing.lex), "f3, e3, f2, e2", PolyRing.lex ); print "Ring: " + str(R); print; [one, Q2, P2, Q1, P1, f3, e3, f2, e2] = R.gens(); print "gens: ", [ str(f) for f in R.gens() ]; print; fp1 = 14 - 12 * e2 - 110 * f2 - 2 * e3 - 10 * f3 - P1; fp2 = 2397 - 2200 * e2 + 240 * f2 - 200 * e3 + 40 * f3 - 20 * Q1; fp3 = 16 * e2**2 - 4 * e2 * e3 - 20 * e2 * f3 + 20 * e3 * f2 + 16 * f2**2 - 4 * f2 * f3 - 12 * e2 + 110 * f2 - P2; fp4 = 2599 * e2**2 - 400 * e2 * e3 + 80 * e2 * f3 - 80 * e3 * f2 + 2599 * f2**2 - 400 * f2 * f3 - 2200 * e2 - 240 * f2 - 20 * Q2; print "fp1: " + str(fp1); print "fp2: " + str(fp2); print "fp3: " + str(fp3); print "fp4: " + str(fp4); print; F = [fp1,fp2,fp3,fp4];
import sys; from jas import PolyRing, QQ, RF from jas import Ideal from jas import startLog from jas import terminate # Montes JSC 2002, 33, 183-208, example 11.3 # integral function coefficients R = PolyRing( PolyRing(QQ(),"r, l, z",PolyRing.lex), "c1, c2, s1, s2", PolyRing.lex ); print "Ring: " + str(R); print; [one,r,l,z,c1,c2,s1,s2] = R.gens(); print "gens: ", [ str(f) for f in r.gens() ]; print; f1 = r - c1 + l * ( s1 * s2 - c1 * c2 ); f2 = z - s1 - l * ( s1 * c2 + s2 * c1 ); f3 = s1**2 + c1**2 - 1; f4 = s2**2 + c2**2 - 1; F = [f1,f2,f3,f4]; print "F: ", [ str(f) for f in F ]; print; startLog();
import sys
from jas import Ring, PolyRing, RF, ZZ
from jas import startLog, terminate
# Hawes & Gibson example 2
# rational function coefficients
#r = Ring( "RatFunc(a, c, b) (y2, y1, z1, z2, x) G" );
r = PolyRing(RF(PolyRing(ZZ(), "a, c, b", PolyRing.lex)), "y2, y1, z1, z2, x",
PolyRing.grad)
print "Ring: " + str(r)
print
[one, a, c, b, y2, y1, z1, z2, x] = r.gens()
p1 = x + 2 * y1 * z1 + 3 * a * y1**2 + 5 * y1**4 + 2 * c * y1
p2 = x + 2 * y2 * z2 + 3 * a * y2**2 + 5 * y2**4 + 2 * c * y2
p3 = 2 * z2 + 6 * a * y2 + 20 * y2**3 + 2 * c
p4 = 3 * z1**2 + y1**2 + b
p5 = 3 * z2**2 + y2**2 + b
p6 = ((p5 / a) / b) / c
print "p6 = ", p6
F = [p1, p2, p3, p4, p5]
g = r.ideal(list=F)
print "Ideal: " + str(g)
print
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])
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)
import sys from jas import Ring, PolyRing, ParamIdeal, QQ, ZM, RR from jas import startLog, terminate # Boolean coefficient boolean GB # see S. Inoue and A. Nagai "On the Implementation of Boolean Groebner Bases" in ASCM 2009 # Z_2 regular ring coefficent example r = PolyRing(RR(ZM(2), 3), "a,x,y", PolyRing.lex) print "r = " + str(r) #print len(r.gens()) [s1, s2, s3, a, x, y] = r.gens() one = r.one() print "one = " + str(one) print "s1 = " + str(s1) print "s2 = " + str(s2) print "s3 = " + str(s3) print "a = " + str(a) print "x = " + str(x) print "y = " + str(y) #brel = [ a**2 - a, x**2 - x, y**2 - y ]; brel = [x**2 - x, y**2 - y] #print "brel = " + str(brel[0]) + ", " + str(brel[1]) + ", " + str(brel[2]); print "brel = " + str(brel[0]) + ", " + str(brel[1]) pl = [(one + s1 + s2) * (x * y + x + y), s1 * x + s1, a * y + a, x * y]
#
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();
import sys; from jas import PolyRing, QQ, RF from jas import Ideal 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; [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 );
import sys
from java.lang import System
from jas import PolyRing, ZM, QQ, RF
from jas import terminate
from jas import startLog
# polynomial examples: factorization over Z_p
p = 5;
cr = PolyRing(ZM(p,field=True),"u",PolyRing.lex );
print "Ring cr: " + str(cr);
[one,u] = cr.gens();
fu = (u**2+u+1)**p;
print "fu = ", fu;
t = System.currentTimeMillis();
G = cr.squarefreeFactors(fu);
t = System.currentTimeMillis() - t;
#print "G = ", G; #.toScript();
print "factor time =", t, "milliseconds";
for h, i in G.iteritems():
print "h**i = (", h, ")**" + str(i);
h = h**i;
print;
qcr = RF(cr);
# $Id$ # import sys from java.lang import System from jas import PolyRing, 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;
q = FF(17, 5)
print "q = " + str(q.ring.toScript())
qe, qa = q.gens()
print "qe = " + str(qe)
print "qa = " + str(qa)
qap = qa**5 - qa + 1
print "qap = " + str(qap)
s = ap - qap
print "s = " + str(s)
ar = PolyRing(r, "beta", PolyRing.lex)
print "ar = " + str(ar)
e, a, b = ar.gens()
print "e = " + str(e)
print "a = " + str(a)
print "b = " + str(b)
p = a**5 - a + beta**5 - (a**3 + a) * beta
#p = p*p;
print "p = " + str(p)
x = p.factors()
print "x = " + ", ".join(
[str(pp) + "**" + str(i) for (pp, i) in x.iteritems()])
g = reduce(operator.mul, [pp**i for (pp, i) in x.iteritems()], e)
print "g = " + str(g)
#print "p-g: " + str(p-g);
print "isFactors(p,x): " + str(p == g)
print
# $Id$ # from java.lang import System from jas import Ring, PolyRing from jas import ZZ, ZM, QQ, AN, RF, CR from jas import terminate, startLog # polynomial examples: factorization #r = Ring( "Z(x) L" ); r = PolyRing(ZZ(), "(x)", PolyRing.lex) print "Ring: " + str(r) print [one, x] = r.gens() #f = x**15 - 1; #f = x * ( x + 1 )**2 * ( x**2 + x + 1 )**3; #f = x**6 - 3 * x**5 + x**4 - 3 * x**3 - x**2 - 3 * x+ 1; #f = x**(3*11*11) + 3 * x**(2*11*11) - x**(11*11); #f = x**(3*11*11*11) + 3 * x**(2*11*11*11) - x**(11*11*11); #f = (x**2+1)*(x-3)*(x-5)**3; #f = x**4 + 1; #f = x**12 + x**9 + x**6 + x**3 + 1; #f = x**24 - 1; #f = x**20 - 1; #f = x**22 - 1; #f = x**8 - 40 * x**6 + 352 * x**4 - 960 * x**2 + 576; #f = 362408718672000 * x**9 + 312179013226080 * x**8 - 591298435728000 * x**6 - 509344705789920 * x**5 - 1178946881112000 * x**2 - 4170783473878580 * x - 2717923400363451;
from basic_sigbased_gb import sigbased_gb 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
# jython examples for jas. # $Id$ # import sys from java.lang import System from jas import QQ, AN, RF, Ring, PolyRing from jas import terminate, startLog # polynomial examples: factorization over Q(sqrt(2))(x)(sqrt(x))[y] Q = PolyRing(QQ(),"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();
from java.lang import Integer from jas import PolyRing, QQ from jas import terminate from jas import startLog # polynomial examples: factorization cr = PolyRing(QQ(),"x",PolyRing.lex ); print "Ring cr: " + str(cr); r = PolyRing(cr,"y,z",PolyRing.lex ); print "Ring: " + str(r); print; [one,x,y,z] = r.gens(); #f = z * ( y + 1 )**2 * ( x**2 + x + 1 )**3; #f = z * ( y + 1 ) * ( x**2 + x + 1 ); #f = ( y + 1 ) * ( x**2 + x + 1 ); #f = ( y + z**2 ) * ( x**2 + x + 1 ); #f = x**4 * y + x**3 + z + x + z**2 + y * z**2; ## f = x**3 + ( ( y + 2 ) * z + 2 * y + 1 ) * x**2 \ ## + ( ( y + 2 ) * z**2 + ( y**2 + 2 * y + 1 ) * z + 2 * y**2 + y ) * x \ ## + ( y + 1 ) * z**3 + ( y + 1 ) * z**2 + ( y**3 + y**2 ) * z + y**3 + y**2; f = ( x + y * z + y + z + 1 ) * ( x**2 + ( y + z ) * x + y**2 + z**2 ); #f = ( x + y * z + y + z + 1 ) * ( x**2 + ( y + z ) * x + y**2 + 1 ); #f = ( x + y ) * ( x - y);
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;
# 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
# $Id$ # 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)
# $Id$ # import sys from java.lang import System from jas import Ring, PolyRing from jas import ZM, QQ, AN, RF, CR from jas import terminate, startLog # polynomial examples: 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) 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) b = i**2 + 1 print "b = " + str(b) c = 1 / i print "c = " + str(c)
from jas import Ring, PolyRing, ParamIdeal, QQ from jas import startLog from jas import terminate # 2 univariate polynomials of degree 2 example for comprehensive GB # integral/rational function coefficients #rp = PolyRing(QQ(), "a,b" ); rp = PolyRing(QQ(), "a" ); r = PolyRing( rp, "x,y,z", PolyRing.lex ); print "Ring: " + str(r); print; #one,a,b,x,y = r.gens(); one,a,x,y,z = r.gens(); #f1 = x * ( x**2 + a * y + b ); #f2 = x * ( y**2 + b * x + a ); f1 = ( x**2 + a * y**2 - x ); f2 = ( a * x**2 + y**2 - y ); f3 = ( x - y ) * z - 1; print "f1 = " + str(f1); print "f2 = " + str(f2); print "f3 = " + str(f3); f = r.paramideal( "", list=[f1,f2,f3] ); print "ParamIdeal: " + str(f); print;
from java.lang import System from jas import PolyRing, QQ from jas import terminate, startLog # polynomial examples: recursive factorization cr = PolyRing(QQ(), "x", PolyRing.lex) print "Ring cr: " + str(cr) print r = PolyRing(cr, "y,z", PolyRing.lex) print "Ring: " + str(r) print [one, x, y, z] = r.gens() #f = z * ( y + 1 )**2 * ( x**2 + x + 1 )**3; #f = z * ( y + 1 ) * ( x**2 + x + 1 ); #f = ( y + 1 ) * ( x**2 + x + 1 ); #f = ( y + z**2 ) * ( x**2 + x + 1 ); #f = x**4 * y + x**3 + z + x + z**2 + y * z**2; ## f = x**3 + ( ( y + 2 ) * z + 2 * y + 1 ) * x**2 \ ## + ( ( y + 2 ) * z**2 + ( y**2 + 2 * y + 1 ) * z + 2 * y**2 + y ) * x \ ## + ( y + 1 ) * z**3 + ( y + 1 ) * z**2 + ( y**3 + y**2 ) * z + y**3 + y**2; f = (x + y * z + y + z + 1) * (x**2 + (y + z) * x + y**2 + z**2) #f = ( x + y * z + y + z + 1 ) * ( x**2 + ( y + z ) * x + y**2 + 1 ); #f = ( x + y ) * ( x - y);
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);
from java.lang import System from java.lang import Integer from jas import Ring, PolyRing from jas import terminate from jas import startLog from jas import QQ, DD # polynomial examples: real roots over Q r = PolyRing(QQ(),"I,x,y,z",PolyRing.lex); print "Ring: " + str(r); print; [one,I,x,y,z] = r.gens(); f1 = z - x - y * I; f2 = I**2 + 1; #f3 = z**3 - 2; f3 = z**3 - 2*I; print "f1 = ", f1; print "f2 = ", f2; print "f3 = ", f3; print; F = r.ideal( list=[f1,f2,f3] ); print "F = ", F;
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()
# # jython examples for jas. # $Id$ # import sys; from jas import PolyRing, Ideal, ZZ from jas import startLog, terminate # tuzun, e-gb example r = PolyRing(ZZ(), "(t)" ); print "Ring: " + str(r); [one,t] = r.gens(); print "one: " + str(one); print "t: " + str(t); print; f1 = 2 * t + 1; f2 = t**2 + 1; F = [f1,f2]; I = r.ideal( list=F ); print "Ideal: " + str(I); print; #startLog(); G = I.eGB();
#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;
#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
import sys; from jas import PolyRing, ZZ, QQ, RealN, CC, ZM, RF, terminate from edu.jas.root import RealArithUtil from edu.jas.arith import ArithUtil # example for rational and real algebraic numbers # # # continued fractions: r = PolyRing(QQ(),"alpha",PolyRing.lex); print "r = " + str(r); e,a = r.gens(); print "e = " + str(e); print "a = " + str(a); sqrt2 = a**2 - 2; print "sqrt2 = " + str(sqrt2); Qs2r = RealN(sqrt2,[1,[3,2]],a-1); #Qs2r = RealN(sqrt2,[-2,-1],a+1); print "Qs2r = " + str(Qs2r.factory()) + " :: " + str(Qs2r.elem); one,alpha = Qs2r.gens(); print "one = " + str(one); print "alpha = " + str(alpha); cf = Qs2r.contFrac(20); print "cf = " + str(cf); nb = Qs2r.contFracApprox(cf);
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);
from jas import terminate
# Montes JSC 2002, 33, 183-208, example 11.2
# integral function coefficients
r = PolyRing(PolyRing(QQ(), "f, e, d, c, b, a", PolyRing.lex), "y,x",
PolyRing.lex)
#r = PolyRing( PolyRing(QQ(),"f, e, d, c, b, a",PolyRing.grad), "y,x", PolyRing.lex );
#r = PolyRing( PolyRing(QQ(),"f, e, d, c, b, a",PolyRing.lex), "y,x", PolyRing.grad );
#r = PolyRing( PolyRing(QQ(),"e, d, c, b, f, a",PolyRing.lex), "y,x", PolyRing.grad );
print "Ring: " + str(r)
print
#[one,e,d,c,b,f,a,y,x] = r.gens();
#automatic: [one,f,e,d,c,b,a,y,x] = r.gens();
print "gens: ", [str(f) for f in r.gens()]
print
f1 = x**2 + b * y**2 + 2 * c * x * y + 2 * d * x + 2 * e * y + f
f2 = x + c * y + d
f3 = b * y + c * x + e
F = [f1, f2, f3]
print "F: ", [str(f) for f in F]
print
#startLog();
If = r.paramideal("", list=F)
print "ParamIdeal: " + str(If)
# $Id$
#
import sys
from java.lang import System
from jas import Ring, PolyRing
from jas import ZM, QQ, AN, RF
from jas import terminate, startLog
# polynomial examples: factorization over Z_p(x)(sqrt{p}(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)
from java.lang import Integer
from jas import PolyRing, QQ, ZM, ZZ
from jas import terminate
from jas import startLog
# polynomial examples: squarefree: characteristic 0
#r = PolyRing(PolyRing(QQ(),"u,v",PolyRing.lex),"x, y",PolyRing.lex)
#r = PolyRing(PolyRing(ZZ(),"u,v",PolyRing.lex),"x, y",PolyRing.lex)
r = PolyRing(PolyRing(ZM(7),"u,v",PolyRing.lex),"x, y",PolyRing.lex)
print "Ring: " + str(r);
print;
[one,u,v,x,y] = r.gens();
a = r.random(k=1,l=3);
b = r.random(k=1,l=3);
c = r.random(k=1,l=3);
if a.isZERO():
a = x;
if b.isZERO():
b = y;
if c.isZERO():
c = y;
f = a**2 * c**3 * b;
print "a = ", a;
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);
print "iq = " + str(iq.factory())
print
iroot = ip.algebraicRoots()
print "algebraic roots: iroot = " + str(iroot)
print "unity algebraic roots: iroot = " + str(iroot.rootsOfUnity())
iroot = iroot.rootRefine(eps)
print "algebraic roots refined: iroot = " + str(iroot.elem.toDecimalScript())
idroot = ip.decimalRoots(eps)
print "decimal roots: idroot = " + str(idroot)
print
r2 = PolyRing(QQ(), "w_3_2", PolyRing.lex)
print "Ring: " + str(r2)
#print;
e, a = r2.gens()
print "e = " + str(e)
print "a = " + str(a)
w3p = (a**3 - 2)
# root{3}(2)
w3q = AN(w3p, 0, True)
print "w3q = " + str(w3q.factory())
print
#w3root = RootFactory.algebraicRoots(w3p.elem);
w3root = w3p.algebraicRoots()
print "algebraic roots: w3root = " + str(w3root)
print "unity algebraic roots: w3root = " + str(w3root.rootsOfUnity())
w3root = w3root.rootRefine(eps)
print "algebraic roots refined: w3root = " + str(w3root.elem.toDecimalScript())
w3droot = w3p.decimalRoots(eps)
cgb = Ipi.CGB()
print "CGB: " + str(cgb)
print
#sys.exit();
# ------------ real roots --------------------
r3 = PolyRing(QQ(), "A", PolyRing.lex)
print "PolyRing: " + str(r3)
print
eps = QQ(1, 10)**DD().elem.DEFAULT_PRECISION
print "eps = ", eps
[one, A] = r3.gens()
plot = {}
plotd = {}
br = 1
for i in range(1, 30):
L = QQ(-1) + (i, 10)
#L = QQ(9,10) + (i,100);
fr = QQ(5) * ((1, 5)) * A**7 - ((21, 20) * L - (29, 20)) * A**6 - (
(21, 10) * L) * A**5 + ((1, 5)) * A**2 - ((1, 20) * L -
(9, 20)) * A - ((1, 10) * L)
print "L = " + str(DD(L))
#print "fr = " + str(fr);
#print;
from java.lang import System from jas import PolyRing, Ideal from jas import terminate, startLog from jas import ZM # system biology examples: GB in Z_2 # see: Informatik Spektrum, 2009, February, # Laubenbacher, Sturmfels: Computer Algebra in der Systembiologie r = PolyRing(ZM(2),"M, B, A, L, P",PolyRing.lex); print "PolyRing: " + str(r); print; [one,M,B,A,L,P] = r.gens(); f1 = M - A; f2 = B - M; f3 = A - A - L * B - A * L * B; f4 = P - M; f5 = L - P - L - L * B - L * P - L * B * P; ## t1 = M - 1; ## t2 = B - 1; ## t3 = A - 1; ## t4 = L - 1; ## t5 = P - 1; # ## t1 = M; ## t2 = B; ## t3 = A;
# jython examples for jas. # $Id$ # import sys from java.lang import System from jas import PolyRing, QQ, AN, RF from jas import terminate, startLog # polynomial examples: prime/primary decomposition in Q(sqrt(2))(x)(sqrt(x))[y,z] Q = PolyRing(QQ(), "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()
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);
from java.lang import System from jas import Ring, PolyRing, QQ, ZM, GF, RF, AN from jas import terminate, startLog, noThreads # polynomial examples: ideal radical decomposition, modified from example 8.16 in GB book # noThreads(); # must be called very early prime = 5 cf = GF(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)
from java.lang import Integer from jas import PolyRing, ZM, QQ from jas import terminate from jas import startLog # polynomial examples: factorization over Z_p r = PolyRing(ZM(1152921504606846883,field=True),"(x)",PolyRing.lex); #r = PolyRing(ZM(19),"x",PolyRing.lex ); #r = PolyRing(ZM(23),"x",PolyRing.lex ); print "Ring: " + str(r); print; [one,x] = r.gens(); #f = x**4 - 1; #f = x**3 + 1; f = x**3 - x - 1; print "f = ", f; print; startLog(); t = System.currentTimeMillis(); #G = r.squarefreeFactors(f); G = r.factorsAbsolute(f);
# import sys; from java.lang import System from jas import Ring, PolyRing from jas import ZM, QQ, AN, RF, CR from jas import terminate, startLog # polynomial examples: complex factorization via algebraic factorization r = PolyRing( CR(QQ()), "x", PolyRing.lex ); print "Ring: " + str(r); print; [one,I,x] = r.gens(); f1 = x**3 - 2; f2 = ( x - I ) * ( x + I ); f3 = ( x**3 - 2 * I ); #f = f1**2 * f2 * f3; f = f1 * f2 * f3; #f = f1 * f2; #f = f1 * f3; #f = f2 * f3; #f = f3; #f = f**3;
# $Id$ # 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)
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\") ---------";
from jas import Ring from jas import PolyRing from jas import Ideal from jas import terminate from jas import startLog from jas import QQ, DD, CR # polynomial examples: complex factorization via algebraic factorization r = PolyRing( CR(QQ()), "x", PolyRing.lex ); print "Ring: " + str(r); print; [one,I,x] = r.gens(); f1 = x**3 - 2; f2 = ( x - I ) * ( x + I ); f3 = ( x**3 - 2 * I ); #f = f1**2 * f2 * f3; f = f1 * f2 * f3; #f = f1 * f2; #f = f1 * f3; #f = f2 * f3; #f = f3;
#
# jython examples for jas.
# $Id$
#
from java.lang import System
from java.lang import Integer
from jas import PolyRing, QQ, ZM, AN
from jas import terminate
from jas import startLog
# polynomial examples: squarefree: characteristic p, finite algebraic
ar = PolyRing(ZM(7,field=True),"i",PolyRing.lex)
[one,i] = ar.gens();
# irred for 7, 11, 19
r = PolyRing(AN(i**2+1,field=True),"x, y, z",PolyRing.lex)
print "Ring: " + str(r);
print;
[one,i,x,y,z] = r.gens();
a = r.random(k=2,l=3);
b = r.random(k=2,l=3);
c = r.random(k=1,l=3);
if a.isZERO():
a = x;
if b.isZERO():
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);
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\") ---------";
import sys; from jas import Ring, PolyRing, ParamIdeal, QQ, ZM, RR from jas import startLog, terminate # Boolean coefficient boolean GB # see S. Inoue and A. Nagai "On the Implementation of Boolean Groebner Bases" in ASCM 2009 # Z_2 regular ring coefficent example r = PolyRing(RR(ZM(2),3),"a,x,y",PolyRing.lex); print "r = " + str(r); #print len(r.gens()) [s1,s2,s3,a,x,y] = r.gens(); one = r.one(); print "one = " + str(one); print "s1 = " + str(s1); print "s2 = " + str(s2); print "s3 = " + str(s3); print "a = " + str(a); print "x = " + str(x); print "y = " + str(y); #brel = [ a**2 - a, x**2 - x, y**2 - y ]; brel = [ x**2 - x, y**2 - y ]; #print "brel = " + str(brel[0]) + ", " + str(brel[1]) + ", " + str(brel[2]); print "brel = " + str(brel[0]) + ", " + str(brel[1]);
# # jython examples for jas. # $Id$ # import sys; from jas import PolyRing, AN, QQ, ZZ, terminate # multiple algebraic field extensions print "------- multiple algebraic field extensions QQ(alpha)(beta)(gamma)(delta) ---------"; r = PolyRing(QQ(),"alpha",PolyRing.lex); print "r = " + str(r); e,a = r.gens(); print "e = " + str(e); print "a = " + str(a); ap = a**5 - a + 1; print "ap = " + str(ap); x = r.factors(ap); print "x = " + str( [ str(p)+"**"+str(e)+"," for (p,e) in x.iteritems() ] ); qs2 = AN(ap,0,len(x)==1); # and e==1 print "qs2 = " + str(qs2.factory()); one,alpha = qs2.gens(); print "one = " + str(one); print "alpha = " + str(alpha); b = alpha**2 - 1; print "b = " + str(b);
