#is automatic: [one,y,z,t] = p.gens(); # no x
#exit(0);
trelations = [t, y, y * t + y,
t, z, z * t - z
];
print "trelations: = " + str([ str(f) for f in trelations ]);
print;
#startLog();
pt = SolvPolyRing(pzc, "t,u", PolyRing.lex, trelations);
print "SolvPolyRing: " + str(pt);
print "sp.gens =" + str([ str(f) for f in pt.gens() ]);
#is automatic: one,y,z,t = rp.gens(); # no x
print;
#exit(0);
a = t**2 + y;
b = t + y + 1;
c = z*t**2 - y * t - z;
print "a = " + str(a);
print "b = " + str(b);
print "c = " + str(c);
#c = c.monic();
#print "c = " + str(c);
print
# 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 ]);
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 I = rp.ideal(list=F) print "SolvableIdeal: " + str(I) print
# + ", 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]) print pt = SolvPolyRing(pzq, "t", PolyRing.lex, trelations) print "SolvPolyRing: " + str(pt) print print "gens =", [str(f) for f in pt.gens()] #print "t = " + str(t); #print "z = " + str(z); zi = 1 / z yi = 1 / y xi = 1 / x #startLog(); a = (t * zi) * y b = t * (zi * y) c = a - b print "t * 1/z * y: " print "a = " + str(a) print "b = " + str(b)
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); print "s4.factory() = " + str(s4.factory()); sr1 = SolvPolyRing(QQ(),"x, y, z",PolyRing.lex,(( z ), ( y ), ( y * z - 1 ),( y ), ( x ), ( x * y - 1 )));
#exit(0);
trelations = [t, y, y * t + y,
t, z, z * t - z
];
print "trelations: = " + str([ str(f) for f in trelations ]);
print;
startLog();
pt = SolvPolyRing(pzc, "t", PolyRing.lex, trelations);
print "SolvPolyRing: " + str(pt);
print;
print "sp.gens =" + str([ str(f) for f in pt.gens() ]);
#is automatic: one,y,z,t = rp.gens(); # no x
#exit(0);
#yi = 1 / y; # not invertible
#print "yi = " + str(yi);
a = t**2 + y;
b = t + y + 1;
c = t**2 - y * t - z;
print "a = " + str(a);
print "b = " + str(b);
print "c = " + str(c);
print
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); print "s4.factory() = " + str(s4.factory()); sr1 = SolvPolyRing(QQ(),"x, y, z",PolyRing.lex,(( z ), ( y ), ( y * z - 1 ),( y ), ( x ), ( x * y - 1 )));
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;
f4 = ( e3**2 * e2**3 + e1 )**3;
#print "f1 = ", f1;
#print "f2 = ", f2;
#print "f3 = ", f3;
#print "f4 = ", f4;
F = [ f1, f2, f3 ];
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);
print;
print "gens = " + str( [ str(r) for r in pt.gens() ] );
#is automatic: one,x,y,z,r,s,t = rp.gens();
#exit(0);
#f1 = u**2 + v**2 + w**2;
#f1 = v;
f1 = w**2 - t;
#print "f1 = " + str(f1);
#f2 = t + u - x**2;
#f2 = f1 * (t - x**2);
#f2 = v**2 - v;
#f2 = v**2 - y;
#f2 = v**2 - z;
f2 = v**2 - t;
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 );
print "SolvableIdeal: " + str(f);
print;
#startLog();
# + ", 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]) print pt = SolvPolyRing(pzq, "t", PolyRing.lex, trelations) print "SolvPolyRing: " + str(pt) print print "gens =", [str(f) for f in pt.gens()] # print "t = " + str(t); # print "z = " + str(z); zi = 1 / z yi = 1 / y xi = 1 / x # startLog(); a = (t * zi) * y b = t * (zi * y) c = a - b print "t * 1/z * y: " print "a = " + str(a) print "b = " + str(b)
## \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;
#startLog();
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)
print "s4.factory() = " + str(s4.factory())
