def __constructJasObject(self):
#from types import StringType
from jas import PolyRing, ZZ
# set up the polynomial ring (Jas syntax)
if 'hasParameters' in self.__dict__ and self.hasParameters != '':
#K = 'K.<%s> = PolynomialRing(ZZ)' % self.hasParameters
#R = K + '; R.<%s> = PolynomialRing(K)' % self.hasVariables
K = PolyRing(ZZ(), str(self.hasParameters))
R = PolyRing(K, str(self.hasVariables))
gens = '%s,%s' % (self.hasParameters, self.hasVariables)
else:
#R = 'R.<%s> = PolynomialRing(ZZ)' % (self.hasVariables)
R = PolyRing(ZZ(), str(self.hasVariables))
gens = str(self.hasVariables)
# translate Jas syntax to pure Python and execute
#exec(preparse(R))
Rg = "(one," + gens + ") = R.gens();"
#print str(R)
exec(str(Rg)) # safe here since R did evaluate
#print "R = " + str(R)
self.jasRing = R
# avoid XSS: check if polynomials are clean
from edu.jas.poly import GenPolynomialTokenizer
vs = GenPolynomialTokenizer.expressionVariables(str(gens))
vs = sorted(vs)
#print "vs = " + str(vs)
vsb = set()
[
vsb.update(GenPolynomialTokenizer.expressionVariables(str(s)))
for s in self.basis
]
vsb = sorted(list(vsb))
#print "vsb = " + str(vsb)
if vs != vsb:
raise ValueError("invalid variables: expected " + str(vs) +
", got " + str(vsb))
# construct polynomials in the constructed ring from
# the polynomial expressions
self.jasBasis = []
for ps in self.basis:
#print "ps = " + str(ps)
ps = str(ps)
ps = ps.replace('^', '**')
#exec(preparse("symbdata_ideal = %s" % ps))
#exec("symbdata_poly = %s" % ps)
pol = eval(ps)
self.jasBasis.append(pol)
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 )');
#
# jython examples for jas.
# $Id$
#
import sys
from jas import PolyRing, QQ, RF
from jas import startLog
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
# $Id$ # 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);
# # jython examples for jas. # $Id$ # 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: 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 jas import PolyRing, ZM, GF from jas import terminate, startLog # system biology examples: GB in Z_2 # see: Informatik Spektrum, 2009, February, # Laubenbacher, Sturmfels: Computer Algebra in der Systembiologie r = PolyRing(GF(2),"M, B, A, L, P",PolyRing.lex); print "PolyRing: " + str(r); print; #automatic: [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; #
import sys from jas import Ring, PolyRing, Ideal, QQ, ZZ, GF, ZM, CC from jas import startLog, terminate # trinks 6/7 example # QQ = rational numbers, ZZ = integers, CC = complex rational numbers, GF = finite field #QQ = QQ(); ZZ = ZZ(); CC = CC(); #r = PolyRing( GF(19),"B,S,T,Z,P,W", PolyRing.lex); #r = PolyRing( GF(1152921504606846883),"B,S,T,Z,P,W", PolyRing.lex); # 2^60-93 #r = PolyRing( GF(2**60-93),"B,S,T,Z,P,W", PolyRing.lex); #r = PolyRing( CC(),"B,S,T,Z,P,W", PolyRing.lex); #r = PolyRing( ZZ(),"B,S,T,Z,P,W", PolyRing.lex); # not for parallel r = PolyRing(QQ(), "B,S,T,Z,P,W", PolyRing.lex) print "Ring: " + str(r) print # sage like: with generators for the polynomial ring #[one,I,B,S,T,Z,P,W] = r.gens(); # is automaticaly included #[one,B,S,T,Z,P,W] = r.gens(); # is automaticaly included f1 = 45 * P + 35 * S - 165 * B - 36 f2 = 35 * P + 40 * Z + 25 * T - 27 * S f3 = 15 * W + 25 * S * P + 30 * Z - 18 * T - 165 * B**2 f4 = -9 * W + 15 * T * P + 20 * S * Z f5 = P * W + 2 * T * Z - 11 * B**3 f6 = 99 * W - 11 * B * S + 3 * B**2 f7 = 10000 * B**2 + 6600 * B + 2673
# # jython examples for jas. # $Id$ # from java.lang import System from jas import GF, PolyRing from jas import terminate, startLog # polynomial examples: factorization over Z_p #r = Ring( "Mod 1152921504606846883 (x) L" ); #r = Ring( "Mod 19 (x) L" ); r = PolyRing( GF(19), "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;
print "KOI = " + str(KOI); o1 = 2 * oneOR + 3 * IOR + 4 * JOR + 5 * KOR + 6 * oneOI + 7 * IOI + 8 * JOI + 9 * KOI; 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;
#
# jruby examples for jas.
# $Id$
#
from java.lang import System
from jas import SolvableRing, SolvPolyRing, PolyRing, RingElem
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;
#startLog();
fl = [ z**2 + y, y**2 + x];
ff = pz.ideal("",fl);
print "ideal ff: " + str(ff);
print;
# jython examples for jas. # $Id$ # import sys from java.lang import System from jas import Ring, PolyRing, QQ, ZM, RF, GF from jas import terminate, startLog, noThreads # polynomial examples: ideal radical decomposition, inseparable cases, dim > 0 #noThreads(); # must be called very early cr = PolyRing(GF(5), "c", PolyRing.lex) print "coefficient Ring: " + str(cr) rf = RF(cr) print "coefficient quotient Ring: " + str(rf.ring) r = PolyRing(rf, "x,y,z", PolyRing.lex) print "Ring: " + str(r) print #automatic: [one,c,x,y,z] = r.gens(); print one, c, x, y, z #sys.exit(); #f1 = (x**2 - 5)**2; #f1 = (y**10 - x**5)**3;
# # jython examples for jas. # $Id$ # import sys from jas import Ring, PolyRing, QQ, ZZ, RingElem from jas import startLog, terminate # GB examples #r = Ring( "Z(t,x,y,z) L" ); #r = Ring( "Mod 11(t,x,y,z) L" ); #r = Ring( "Rat(t,x,y) L" ); r = PolyRing(QQ(), "t,x,y", PolyRing.lex) #r = PolyRing( ZZ(), "t,x,y", PolyRing.lex ); print "Ring: " + str(r) print ps = """ ( ( t - x - 2 y ), ( x^4 + 4 ), ( y^4 + 4 ) ) """ # ( y^4 + 4 x y^3 - 2 y^3 - 6 x y^2 + 1 y^2 + 6 x y + 2 y - 2 x + 3 ), # ( x^2 + 1 ), # ( y^3 - 1 )
# # jython examples for jas. # $Id$ # import sys from jas import Ring, PolyRing, Ideal, PSIdeal, QQ, ZM from jas import startLog, terminate # example from CLO(UAG), 4.4 #R = PolyRing( GF(32003), "x,y,z" ); R = PolyRing(QQ(), "x,y,z") print "Ring: " + str(R) print #automatic: [one,x,y,z] = R.gens(); f1 = x**5 - x * y**6 - z**7 f2 = x * y + y**3 + z**3 f3 = x**2 + y**2 - z**2 L = [f1, f2, f3] #print "L = ", str(L); F = R.ideal(list=L) print "Ideal: " + str(F) print PR = R.powerseriesRing()
# # jython examples for jas. # $Id$ # import sys from java.lang import System from jas import Ring, RF, QQ, PolyRing from jas import terminate, startLog # elementary integration r = PolyRing(QQ(), "x", PolyRing.lex) print "r = " + str(r) rf = RF(r) print "rf = " + str(rf.factory()) [one, x] = rf.gens() print "one = " + str(one) print "x = " + str(x) print #f = 1 / ( 1 + x**2 ); #f = x**2 / ( x**2 + 1 ); #f = 1 / ( x**2 - 2 ); #f = 1 / ( x**3 - 2 ); #f = ( x + 3 ) / ( x**2- 3 * x - 40 );
## 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 = WordPolyRing(PolyRing(ZZ(), "a, c, b", PolyRing.lex), "y2, y1, z1, z2, x")
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]
# make all variables commute
cm = [RingElem(q) for q in r.ring.commute()]
# # 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);
F = r.ideal(list=[r1, r2, r3, r4, r5, r6, r7, r8, f1, f2, f3]) #F = r.ideal( list=[r1,r2,f1,f2,f3] ); print "F = " + str(F) print startLog() G = F.GB() 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();
## 3 z_1^2+y_1^2+b \&
## 3z_2^2+y_2^2+b \&
## \end{Equations}
## \end{PossoExample}
import sys;
from jas import Ring, PolyRing, QQ
from jas import startLog, terminate
# Hawes & Gibson example 2
# rational function coefficients
#r = Ring( "IntFunc(a, c, b) (y2, y1, z1, z2, x) G" );
r = PolyRing( PolyRing(QQ(),"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( list=F );
print "Ideal: " + str(g);
## 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}
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]
# # jython examples for jas. # $Id$ # import sys from java.lang import System from jas import PolyRing, QQ, DD, AN, BigDecimal from jas import terminate, startLog # roots simplification 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)
# # jython examples for jas. # $Id$ # import sys from java.lang import System from jas import Ring, PolyRing, QQ, DD from jas import terminate, startLog # polynomial examples: real roots over Q r = PolyRing(QQ(), "I,x,y,z", PolyRing.lex) print "Ring: " + str(r) print #automatic: [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
# jython examples for jas.
# $Id$
#
import sys;
from jas import PolyRing, QQ
from jas import startLog, terminate
# Raksanyi & Walter example
# integral/rational function coefficients
#r = Ring( "RatFunc(a1, a2, a3, a4) (x1, x2, x3, x4) L" );
#r = Ring( "IntFunc(a1, a2, a3, a4) (x1, x2, x3, x4) L" );
r = PolyRing( PolyRing(QQ(),"a1, a2, a3, a4",PolyRing.lex),
"x1, x2, x3, x4", PolyRing.lex);
print "Ring: " + str(r);
print;
ps = """
(
( 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 = r.paramideal( ps );
print "ParamIdeal: " + str(f);
print;
#
# jython examples for jas.
# $Id$
#
from java.lang import System
from jas import PolyRing, QQ, ZM
from jas import terminate, startLog
import operator
# polynomial examples: squarefree: characteristic 0
r = PolyRing(QQ(), "x, y, z", PolyRing.lex)
print "Ring: " + str(r)
print
#automatic: [one,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
#
# jython examples for jas.
# $Id$
#
import sys
from jas import QQ, PolyRing
from jas import startLog
from jas import terminate
# ideal elimination example
#r = Ring( "Rat(x,y,z) G" );
r = PolyRing(QQ(), "(x,y,z)", PolyRing.grad)
print "Ring: " + str(r)
print
ps1 = """
(
( x^2 - 2 ),
( y^2 - 3 ),
( z^3 - x * y )
)
"""
ff = [x**2 - 2, y**2 - 3, z**3 - x * y]
#F1 = r.ideal( ps1 );
F1 = r.ideal("", ff)
print "Ideal: " + str(F1)
import sys
from jas import Ring, PolyRing, QQ, ZZ, Order, Scripting
from jas import startLog, terminate
# examples: from Mathematica
# check("GroebnerBasis({x^2 + y^2 + z^2 - 1, x - z + 2, z^2 - x*y},
# {x, y, z})",
# "{12-28*z+27*z^2-12*z^3+3*z^4, -6+4*y+11*z-6*z^2+3*z^3, 2+x-z}");
Scripting.setCAS(Scripting.CAS.Math)
#Scripting.setCAS(Scripting.CAS.Sage);
#Scripting.setCAS(Scripting.CAS.Singular);
r = PolyRing(ZZ(), "x,y,z", Order.Lexicographic)
print "Ring: " + str(r)
print
ff = [x**2 + y**2 + z**2 - 1, x - z + 2, z**2 - x * y]
F = r.ideal("", ff)
print "F = " + str(F)
print
#startLog();
G = F.GB()
print "G = " + str(G)
print
print "Ma: " + str(
# # jython examples for jas. # $Id$ # 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")
# # jython examples for jas. # $Id$ # import sys from jas import PolyRing, QQ, RF from jas import startLog from jas import terminate # Montes JSC 2002, 33, 183-208, example 5.1 # integral function coefficients r = PolyRing(PolyRing(QQ(), "a, b", PolyRing.lex), "u,z,y,x", PolyRing.lex) print "Ring: " + str(r) print #automatic: [one,a,b,u,z,y,x] = r.gens(); print "gens: ", [str(f) for f in r.gens()] print f1 = a * x + 2 * y + 3 * z + u - 6 f2 = x + 3 * y - z + 2 * u - b f3 = 3 * x - a * y + z - 2 f4 = 5 * x + 4 * y + 3 * z + 3 * u - 9 F = [f1, f2, f3, f4] print "F: ", [str(f) for f in F] print
# jython examples for jas. # $Id$ # import sys; 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: complex roots over Q for zero dimensional ideal `cyclic5' r = PolyRing(QQ(),"a,b,c,d,e",PolyRing.lex); print "Ring: " + str(r); print; #is automatic: [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;
#
# jython examples for jas.
# $Id$
#
import sys
from jas import Ring, PolyRing, RF, ZZ
from jas import terminate
# Raksanyi & Walter example
# rational function coefficients
#r = Ring( "RatFunc(a1, a2, a3, a4) (x1, x2, x3, x4) G" );
r = PolyRing(RF(PolyRing(ZZ(), "a1, a2, a3, a4", PolyRing.lex)),
"x1, x2, x3, x4", PolyRing.grad)
print "Ring: " + str(r)
print
ps = """
(
( 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 = r.ideal(ps)
print "Ideal: " + str(f)
print
