print "Ring: " + str(r) #print; #automatic: [one,I] = r.gens(); print "one = ", one print "I = ", I eps = QQ(1, 10)**(BigDecimal.DEFAULT_PRECISION) #-3 #eps = QQ(1,10) ** 7; #eps = nil; print "eps = " + str(eps) ip = (I**2 + 1) # I iq = AN(ip, 0, True) 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;
print "s3 = " + str(s3); s4 = s3 - s1; print "s4 = " + str(s4); print "s4.factory() = " + str(s4.factory()); print; print "------- AN(alpha**2 - 2) ---------"; 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); Qs2 = AN(sqrt2); print "Qs2 = " + str(Qs2.factory()); [one,alpha] = Qs2.gens(); print "one = " + str(one); print "alpha = " + str(alpha); b = alpha**2 - 2; print "b = " + str(b); c = 1 / alpha; print "c = " + str(c); Qs2x = AN(alpha**2 - 2); print "Qs2x = " + str(Qs2x.factory()); print; print "------- GF_17(alpha**2 - 2) ---------"; r = PolyRing(GF(17),"alpha",PolyRing.lex);
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(); #print "ep = " + str(ep); #print "wp = " + str(wp); #print "ap = " + str(ap); print; Qr = RF(Qp);
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) print "aa2 = " + str(aa2) print #cr = PolyRing(QQ(),"t",PolyRing.lex); cr = PolyRing(Q3a, "t", PolyRing.lex) print "coefficient Ring: " + str(cr) rf = RF(cr) print "coefficient quotient Ring: " + str(rf.ring.toScript())
print "s3 = " + str(s3); s4 = s3 - s1; print "s4 = " + str(s4); print "s4.factory() = " + str(s4.factory()); print; print "------- AN(alpha**2 - 2) ---------"; 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); Qs2 = AN(sqrt2); print "Qs2 = " + str(Qs2.factory()); [one,alpha] = Qs2.gens(); print "one = " + str(one); print "alpha = " + str(alpha); b = alpha**2 - 2; print "b = " + str(b); c = 1 / alpha; print "c = " + str(c); Qs2x = AN(alpha**2 - 2); print "Qs2x = " + str(Qs2x.factory()); print; print "------- GF_17(alpha**2 - 2) ---------"; r = PolyRing(GF(17),"alpha",PolyRing.lex);
r = PolyRing( QQ(), "I", PolyRing.lex ); print "Ring: " + str(r); #print; #automatic: [one,I] = r.gens(); print "one = ", one; print "I = ", I; eps = QQ(1,10) ** (BigDecimal.DEFAULT_PRECISION); #-3 #eps = QQ(1,10) ** 7; #eps = nil; print "eps = " + str(eps); ip = (I**2 + 1); # I iq = AN( ip, 0, True); 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);
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 "aa2 = " + str(aa2); print; #cr = PolyRing(QQ(),"t",PolyRing.lex); cr = PolyRing(Q3a,"t",PolyRing.lex); print "coefficient Ring: " + str(cr); rf = RF(cr); print "coefficient quotient Ring: " + str(rf.ring.toScript());
[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); [et2,t,wpt] = cr.gens(); print "et2 = " + str(et2); print "t = " + str(t); print "wpt = " + str(wpt); print; root = wpt**prime - ta2; af = AN(root,field=True); print "coefficient algebraic quotient ring: " + str(af.ring.toScript()); #print af.gens(); ##xx = AN(( wpt**5 + 4 * t ),True,PolyRing(RF(PolyRing(ZM(5),"t",PolyRing.lex)),"wpt",PolyRing.lex)) ##print "xx: " + str(xx.ring.toScript()); [one,t,wpt] = af.gens(); print "one = " + str(one); print "t = " + str(t); print "wpt = " + str(wpt); #print one,t,wpt; print; #sys.exit(); r = PolyRing(af,"x,y",PolyRing.lex);
from jas import Ring from jas import PolyRing from jas import Ideal from jas import QQ, AN from jas import terminate from jas import startLog # polynomial examples: factorization over Q(i)(sqrt2) Q = PolyRing(QQ(),"i",PolyRing.lex); print "Q = " + str(Q); Q.inject_variables(); print "i = " + str(i); imag = i**2 + 1; print "imag = " + str(imag); Qi = AN(imag,field=True); print "Qi = " + str(Qi.factory()); Qi.inject_variables(); print "i = " + str(i); print; Wr = PolyRing(Qi,"w2",PolyRing.lex) print "Wr = " + str(Wr); Wr.inject_variables(); print "i = " + str(i); print "w2 = " + str(w2); w2p = w2**2 - 2; print "w2p = " + str(w2p); Qw2 = AN(w2p,field=True); print "Qw2 = " + str(Qw2.factory()); Qw2.inject_variables();
from java.lang import System from jas import QQ, AN, Ring, PolyRing from jas import terminate, startLog # polynomial examples: factorization over Q(i)(sqrt2) Q = PolyRing(QQ(), "i", PolyRing.lex) print "Q = " + str(Q) # is automatic: Q.inject_variables(); print "i = " + str(i) imag = i ** 2 + 1 print "imag = " + str(imag) Qi = AN(imag, field=True) print "Qi = " + str(Qi.factory()) # is automatic: Qi.inject_variables(); print "i = " + str(i) print Wr = PolyRing(Qi, "w2", PolyRing.lex) print "Wr = " + str(Wr) # is automatic: Wr.inject_variables(); print "i = " + str(i) print "w2 = " + str(w2) w2p = w2 ** 2 - 2 print "w2p = " + str(w2p) Qw2 = AN(w2p, field=True) print "Qw2 = " + str(Qw2.factory())
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) #is automatic: [one,i,x] = r.gens(); print "one = " + str(one) print "i = " + str(i) print "x = " + str(x) print #f = x**7 - 1;
# jython examples for jas.
# $Id$
#
from java.lang import System
from jas import PolyRing, QQ, ZM, AN, GF
from jas import terminate, startLog
# polynomial examples: squarefree: characteristic p, finite algebraic
ar = PolyRing(GF(7), "i", PolyRing.lex)
#automatic: [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
#automatic: [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():
b = y
if c.isZERO():
c = z
[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) [et2, t, wpt] = cr.gens() print "et2 = " + str(et2) print "t = " + str(t) print "wpt = " + str(wpt) print root = wpt**prime - ta2 af = AN(root, field=True) print "coefficient algebraic quotient ring: " + str(af.ring.toScript()) #print af.gens(); ##xx = AN(( wpt**5 + 4 * t ),True,PolyRing(RF(PolyRing(ZM(5),"t",PolyRing.lex)),"wpt",PolyRing.lex)) ##print "xx: " + str(xx.ring.toScript()); [one, t, wpt] = af.gens() print "one = " + str(one) print "t = " + str(t) print "wpt = " + str(wpt) #print one,t,wpt; print #sys.exit(); r = PolyRing(af, "x,y", PolyRing.lex)
from java.lang import System from jas import QQ, AN, Ring, PolyRing from jas import terminate, startLog # polynomial examples: factorization over Q(i)(sqrt2) Q = PolyRing(QQ(), "i", PolyRing.lex) print "Q = " + str(Q) #is automatic: Q.inject_variables(); print "i = " + str(i) imag = i**2 + 1 print "imag = " + str(imag) Qi = AN(imag, field=True) print "Qi = " + str(Qi.factory()) #is automatic: Qi.inject_variables(); print "i = " + str(i) print Wr = PolyRing(Qi, "w2", PolyRing.lex) print "Wr = " + str(Wr) #is automatic: Wr.inject_variables(); print "i = " + str(i) print "w2 = " + str(w2) w2p = w2**2 - 2 print "w2p = " + str(w2p) Qw2 = AN(w2p, field=True) print "Qw2 = " + str(Qw2.factory())
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) # is automatic: [one,i,x] = r.gens(); print "one = " + str(one) print "i = " + str(i) print "x = " + str(x) print
Qwx = PolyRing(Qr, "wx", PolyRing.lex) print "Qwx = " + str(Qwx) [ewx, ax, wx] = Qwx.gens() #print "ewx = " + str(ewx); print "ax = " + str(ax) print "wx = " + str(wx) print #rootx = wx**5 - 2; # not working #rootx = wx**5 - 1/ax; #rootx = wx**5 - ax; #rootx = wx**5 - ( ax**2 + 3 * ax ); rootx = wx**5 - (ax**2 + 3 / ax) print "rootx = " + str(rootx) Q2x = AN(rootx, field=True) print "Q2x = " + str(Q2x.factory()) [ex2, ax2, wx] = Q2x.gens() #print "ex2 = " + str(ex2); #print "w2x2 = " + str(w2x2); #print "ax2 = " + str(ax2); #print "wx = " + str(wx); print #Yr = PolyRing(Q2x,"y,z,c0,c1,c2",PolyRing.lex) Yr = PolyRing(Q2x, "y,z", PolyRing.lex) print "Yr = " + str(Yr) #[e,x,wx,y,z,c0,c1,c2] = Yr.gens(); [e, x, wx, y, z] = Yr.gens() print "e = " + str(e)
Qwx = PolyRing(Qr,"wx",PolyRing.lex); print "Qwx = " + str(Qwx); [ewx,ax,wx] = Qwx.gens(); #print "ewx = " + str(ewx); print "ax = " + str(ax); print "wx = " + str(wx); print; #rootx = wx**5 - 2; # not working #rootx = wx**5 - 1/ax; #rootx = wx**5 - ax; #rootx = wx**5 - ( ax**2 + 3 * ax ); rootx = wx**5 - ( ax**2 + 3 / ax ); print "rootx = " + str(rootx); Q2x = AN(rootx,field=True); print "Q2x = " + str(Q2x.factory()); [ex2,ax2,wx] = Q2x.gens(); #print "ex2 = " + str(ex2); #print "w2x2 = " + str(w2x2); #print "ax2 = " + str(ax2); #print "wx = " + str(wx); print; #Yr = PolyRing(Q2x,"y,z,c0,c1,c2",PolyRing.lex) Yr = PolyRing(Q2x,"y,z",PolyRing.lex) print "Yr = " + str(Yr); #[e,x,wx,y,z,c0,c1,c2] = Yr.gens(); [e,x,wx,y,z] = Yr.gens(); print "e = " + str(e);
# 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); c = 1 / b; print "c = " + str(c); print "b*c = " + str(b*c); print ar = PolyRing(qs2,"beta",PolyRing.lex); print "ar = " + str(ar); e,a,b = ar.gens();
