print "G = " + str(G) print "isGB(G) = " + str(G.isGB()) print # now as solvable polynomials p = PolyRing(QQ(), "a,b,e1,e2,e3") #is automatic: [one,a,b,e1,e2,e3] = p.gens(); relations = [e3, e1, e1 * e3 - e1, e3, e2, e2 * e3 - e2] print "relations: =", [str(f) for f in relations] print #rp = SolvPolyRing(QQ(), "a,b,e1,e2,e3", rel=relations); rp = SolvPolyRing(QQ(), "a,b,e1,e2,e3", PolyRing.lex, relations) print "SolvPolyRing: " + str(rp) print print "gens =", [str(f) for f in rp.gens()] #[one,a,b,e1,e2,e3] = rp.gens(); #[one,I,J,K,a,b,e1,e2,e3] = rp.gens(); f1 = e1 * e3**3 + e2**10 - a f2 = e1**3 * e2**2 + e3 f3 = e3**3 + e3**2 - b F = [f1, f2, f3] print "F =", [str(f) for f in F] print
from java.lang import System from jas import SolvableRing, SolvPolyRing, PolyRing from jas import QQ, startLog, SRC, SRF # Ore extension solvable polynomial example, Gomez-Torrecillas, 2003 pcz = PolyRing(QQ(), "x,y,z") #is automatic: [one,x,y,z] = pcz.gens(); zrelations = [z, y, y * z + x] print "zrelations: = " + str([str(f) for f in zrelations]) print pz = SolvPolyRing(QQ(), "x,y,z", 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(f) for f in pzq.ring.generators() ]); print pct = PolyRing(pzq, "t") #is automatic: [one,x,y,z,t] = p.gens(); trelations = [t, y, y * t + y, t, z, z * t - z] print "relations: = " + str([str(f) for f in trelations])
# Ore extension solvable polynomial example, Gomez-Torrecillas, 2003
pcz = PolyRing(QQ(),"x,y,z,t", PolyRing.lex);
#is automatic: [one,x,y,z,t] = p.gens();
zrelations = [z, y, y * z + x,
t, y, y * t + y,
t, z, z * t - z
];
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,
for g in r.gens():
print "g = " + str(g);
print;
print "------- SolvPolyRing(QQ(),\"x,y,z\") ---------";
r = PolyRing(QQ(),"x,y,z",PolyRing.lex);
print "r = " + str(r);
[pone,px,py,pz] = r.gens();
print "pone = " + str(pone);
print "px = " + str(px);
print "py = " + str(py);
print "pz = " + str(pz);
rel = ( py, px, px * py - 1 , pz, py, py * pz - 1 );
#print "rel = " + str(rel);
sr = SolvPolyRing(QQ(),"x,y,z",PolyRing.lex,rel);
print "sr = " + str(sr);
[one,x,y,z] = sr.gens();
print "one = " + str(one);
print "x = " + str(x);
print "y = " + str(y);
print "z = " + str(z);
print "one.factory() = " + str(one.factory());
s1 = QQ(1,2) + QQ(2,3) * x + QQ(2,5) * y + ( x + y + z )**2;
print "s1 = " + str(s1);
s2 = (1,2) + (2,3) * x + (2,5) * y + ( x + y + z )**2;
print "s2 = " + str(s2);
s3 = z**2 + 2 * y * z + 2 * x * z + y**2 + 2 * x * y + x**2 + (2,5) * y + (2,3) * x + (1,2);
print "s3 = " + str(s3);
s4 = s1 - s3;
print "s4 = " + str(s4);
import sys from jas import SolvableRing, SolvPolyRing, PolyRing, SolvableIdeal from jas import QQ, ZZ, GF, SRF, startLog, terminate # Iterated Ore extension solvable polynomial example, rc = PolyRing(QQ(), "x,y,z,t", PolyRing.lex) #is automatic: [one,x,y,z,t] = rc.gens(); crel = [z, y, y * z + x, t, y, y * t + y, t, z, z * t - z] print "crel: = " + str([str(r) for r in crel]) print rcs = SolvPolyRing(QQ(), "x,y,z,t", PolyRing.lex, crel) #exit(0) rm = PolyRing(rcs, "u,v,w", PolyRing.lex) #is automatic: [one,x,y,z,t,u,v,w] = rm.gens(); mrel = [v, u, u * v + x, w, v, v * w + y, w, u, u * w - z] print "mrel: = " + str([str(r) for r in mrel]) print rs = SolvPolyRing(rcs, "u,v,w", PolyRing.lex, mrel) #exit(0)
RF(PolyRing(coeff,"b1,c1",PolyRing.lex)),
"E,D1,D2,D3",PolyRing.grad);
print "PolyRing: " + str(p);
print;
relations = [
( D3 ), ( D1 ), ( D1 * D3 - c1 * E + b1 * E ),
( D3 ), ( D2 ), ( D2 * D3 - E + E ),
( D2 ), ( D1 ), ( D1 * D2 - c1 * E + b1 * E )
];
print "relations: = " + str([ str(f) for f in relations ]);
print;
rp = SolvPolyRing(
RF(PolyRing(coeff,"b1,c1",PolyRing.lex)),
"E,D1,D2,D3",PolyRing.grad,relations);
print "SolvPolyRing: rp = " + str(rp);
print;
print "gens = " + str([ str(f) for f in rp.gens() ]);
print;
F = [
( D1 - D2 * D3 ),
( D2 * D3 + D1 ),
( - D1 * D3 + D2 )
];
print "F =" + str([ str(f) for f in F ]);
print
f = rp.ideal( list=F );
from java.lang import System
from jas import SolvableRing, SolvPolyRing, PolyRing, SRF
from jas import QQ, startLog, SolvableModule, SolvableSubModule, terminate
# Ore extension solvable polynomial example, modules
rp = PolyRing(QQ(), "x,y,z,t", PolyRing.lex)
#is automatic: [one,x,y,z,t] = rp.gens();
trel = [z, y, y * z + x, t, y, y * t + y, t, z, z * t - z]
print "trel: = " + str([str(f) for f in trel])
print
rs = SolvPolyRing(QQ(), "x,y,z,t", PolyRing.lex, trel)
#exit(0)
f = rs.ideal("", [t**2 + z**2 + x**2 + y**2 + 1])
print "f: " + str(f)
tf = f.twosidedGB()
print "t: " + str(tf)
print
#exit(0)
r = SolvableModule("", rs)
print "SolvableModule: " + str(r)
print
## 2 z_2+6ay_2+20 y_2^3+2c \&
## 3 z_1^2+y_1^2+b \&
## 3z_2^2+y_2^2+b \&
## \end{Equations}
## \end{PossoExample}
#from java.lang import System
from jas import WordRing, WordPolyRing, WordPolyIdeal, PolyRing, SolvPolyRing, RingElem
from jas import terminate, startLog
from jas import QQ, ZZ, GF, ZM, WRC
# Hawes & Gibson example 2
# rational function coefficients
r = SolvPolyRing( 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;
F = [p1,p2,p3,p4,p5];
g = r.ideal( "", F );
print "Ideal: " + str(g);
print "r = " + str(r)
for g in r.gens():
print "g = " + str(g)
print
print "------- SolvPolyRing(QQ(),\"x,y,z\") ---------"
r = PolyRing(QQ(), "x,y,z", PolyRing.lex)
print "r = " + str(r)
[pone, px, py, pz] = r.gens()
print "pone = " + str(pone)
print "px = " + str(px)
print "py = " + str(py)
print "pz = " + str(pz)
rel = (py, px, px * py - 1, pz, py, py * pz - 1)
#print "rel = " + str(rel);
sr = SolvPolyRing(QQ(), "x,y,z", PolyRing.lex, rel)
print "sr = " + str(sr)
[one, x, y, z] = sr.gens()
print "one = " + str(one)
print "x = " + str(x)
print "y = " + str(y)
print "z = " + str(z)
print "one.factory() = " + str(one.factory())
s1 = QQ(1, 2) + QQ(2, 3) * x + QQ(2, 5) * y + (x + y + z)**2
print "s1 = " + str(s1)
s2 = (1, 2) + (2, 3) * x + (2, 5) * y + (x + y + z)**2
print "s2 = " + str(s2)
s3 = z**2 + 2 * y * z + 2 * x * z + y**2 + 2 * x * y + x**2 + (2, 5) * y + (
2, 3) * x + (1, 2)
print "s3 = " + str(s3)
s4 = s1 - s3
#from java.lang import Integer from jas import SolvableRing, SolvPolyRing, PolyRing from jas import QQ, startLog, SRC, SRF # Weyl coefficient field example r = PolyRing(QQ(), "p1,q1") #is automatic: [one,p1,q1] = p.gens(); relations = [q1, p1, p1 * q1 + 1] print "relations: = " + str([str(f) for f in relations]) print rp = SolvPolyRing(QQ(), "p1,q1", PolyRing.lex, relations) print "SolvPolyRing: " + str(rp) print print "gens =" + str([str(f) for f in rp.gens()]) #is automatic: one,p1,q1 = rp.gens(); scp = SRF(rp) print "scp = " + str(scp) r2 = PolyRing(scp, "p2,q2") #is automatic: [one,p1,q1,p2,q2] = r2.gens(); relations2 = [q2, p2, p2 * q2 + 1] print "relations: = " + str([str(f) for f in relations2])
from jas import terminate
# simple example for solvable comprehensive GB
# integral/rational function coefficients
rc = PolyRing( PolyRing(QQ(),"a,b",PolyRing.lex),"x,d", PolyRing.lex );
print "commutativ Ring: " + str(rc);
print;
rel = [d, x, x * d + 1
];
print "relations: = " + str([ str(f) for f in rel ]);
print;
r = SolvPolyRing( PolyRing(QQ(),"a,b",PolyRing.lex),"x,d", PolyRing.lex, rel);
print "Ring: " + str(r);
print;
p1 = 2 * x * d**2 + a * x * d;
p2 = x * d**3 + b * x**2 * d - b * x;
p3 = x * d**2 - a * x;
f = r.paramideal( "", [p1,p2,p3] );
print "ParamIdeal: " + str(f);
print;
#exit();
#startLog();
gs = f.CGBsystem();
from jas import startLog from jas import terminate # simple example for comprehensive GB # integral/rational function coefficients rc = PolyRing( PolyRing(QQ(),"(u,v)",PolyRing.lex),"(x,y)", PolyRing.lex ); print "comm Ring: " + str(rc); print; rel = [y, x, x * y + 1]; print "relations: = " + str([ str(f) for f in rel ]); print; r = SolvPolyRing( PolyRing(QQ(),"(u,v)",PolyRing.lex),"(x,y)", PolyRing.lex, rel); print "Ring: " + str(r); print; p1 = v * x * y + x; p2 = u * y**2 + x**2; f = r.paramideal( "", [p1,p2] ); print "ParamIdeal: " + str(f); print; #sys.exit(); #startLog();
import sys
from jas import PolyRing, SolvPolyRing, QQ
from jas import startLog, terminate
# Ore extension solvable polynomial example, Gomez-Torrecillas, 2003
p = PolyRing(QQ(), "x,y,z,t")
#is automatic: [one,x,y,z,t] = p.gens();
relations = [z, y, y * z + x, t, y, y * t + y, t, z, z * t - z]
print "relations: = " + str([str(f) for f in relations])
print
rp = SolvPolyRing(QQ(), "x,y,z,t", PolyRing.lex, relations)
print "SolvPolyRing: " + str(rp)
print
print "gens =" + str([str(f) for f in rp.gens()])
#is automatic: one,x,y,z,t = rp.gens();
#f1 = x**2 + y**2 + z**2 + t**2 + 1;
#print "f1 = " +str(f1);
ff = [x, y, z, t**2 - 1]
print "ff = " + str([str(f) for f in ff])
print
ii = rp.ideal("", ff)
print "SolvableIdeal: " + str(ii)
