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 )');
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)
#
# 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)
#
# 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
]
# # 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. # $Id$ # 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, RF, ZZ, QQ
from jas import startLog, terminate
from edu.jas.arith import ModIntegerRing
#startLog();
# 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;
ps = """
(
( x + 2 y1 z1 + { 3 a } y1^2 + 5 y1^4 + { 2 c } y1 ),
( x + 2 y2 z2 + { 3 a } y2^2 + 5 y2^4 + { 2 c } y2 ),
( 2 z2 + { 6 a } y2 + 20 y2^3 + { 2 c } ),
( 3 z1^2 + y1^2 + { b } ),
( 3 z2^2 + y2^2 + { b } )
)
""";
f = r.paramideal( ps );
print "Ideal: " + str(f);
import sys 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();
# # 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
# $Id$ # from java.lang import System 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 = 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();
## 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]
#
# 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. # $Id$ # from java.lang import System from jas import Ring, PolyRing, QQ, DD from jas import terminate, startLog # polynomial examples: ideal prime decomposition #r = Ring( "Rat(x) L" ); #r = Ring( "Q(x) L" ); r = PolyRing(QQ(), "x,y,z", PolyRing.lex) print "Ring: " + str(r) print #automatic: [one,x,y,z] = r.gens(); f1 = (x**2 - 5)**2 f2 = y**3 - x f3 = z**2 - y * x #print "f1 = ", f1; print "f2 = ", f2 print "f3 = ", f3 print F = r.ideal(list=[f2, f3])
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;
# jython examples for jas. # $Id$ # 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;
# # 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")
## 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) L" );
r = PolyRing(RF(PolyRing(ZZ(), "a, c, b", PolyRing.lex)), "y2, y1, z1, z2, x",
PolyRing.lex)
print "Ring: " + str(r)
print
ps = """
(
( x + 2 y1 z1 + { 3 a } y1^2 + 5 y1^4 + { 2 c } y1 ),
( x + 2 y2 z2 + { 3 a } y2^2 + 5 y2^4 + { 2 c } y2 ),
( 2 z2 + { 6 a } y2 + 20 y2^3 + { 2 c } ),
( 3 z1^2 + y1^2 + { b } ),
( 3 z2^2 + y2^2 + { b } )
)
"""
f = r.ideal(ps)
print "Ideal: " + str(f)
#
# 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
# # jython examples for jas. # $Id$ # 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
# $Id$ # 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 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;
#
# 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
# 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.
# $Id$
#
from java.lang import System
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():
# jython examples for jas.
# $Id$
#
import sys;
from jas import Ring, PolyRing, Ideal
from jas import QQ, ZZ, RF
from jas import startLog, terminate
# example: Algebraic Statistics
# Drton, Sturmfels, Sullivant, example 2.1.3
r = PolyRing(RF(PolyRing(QQ(),"u0,u1,u2,u12",PolyRing.lex)),"l1,l2",PolyRing.grad);
print "Ring: " + str(r);
print;
print one,u0,u1,u2,u12,l1,l2;
f1 = (u1+u12)*(l1+l2+2)*(l1+1)*(l1+l2+1)\
+ (u12)*l1*(l1+1)*(l1+l2+1)\
- (u2+u12)*l1*(l1+l2+2)*(l1+l2+1)\
- (u0+u1+u2+u12)*l1*(l1+l2+2)*(l1+1) ;
f2 = (u2+u12)*(l1+l2+2)*(l2+1)*(l1+l2+1)\
+ (u12)*l2*(l2+1)*(l1+l2+1)\
- (u1+u12)*l2*(l1+l2+2)*(l1+l2+1)\
- (u0+u1+u2+u12)*l2*(l1+l2+2)*(l2+1) ;
# # jython examples for jas. # $Id$ # 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;
import sys; from jas import QQ, Ring, PolyRing, Ideal from jas import startLog, terminate # trinks 6/7 example #r = Ring( "Mod 19 (B,S,T,Z,P,W) L" ); #r = Ring( "Mod 1152921504606846883 (B,S,T,Z,P,W) L" ); # 2^60-93 #r = Ring( "Quat(B,S,T,Z,P,W) L" ); #r = Ring( "Z(B,S,T,Z,P,W) L" ); #r = Ring( "C(B,S,T,Z,P,W) L" ); #r = Ring( "Rat(B,S,T,Z,P,W) L" ); r = PolyRing( QQ(),"B,S,T,Z,P,W", PolyRing.lex); print "Ring: " + str(r); print; ps = """ ( ( 45 P + 35 S - 165 B - 36 ), ( 35 P + 40 Z + 25 T - 27 S ), ( 15 W + 25 S P + 30 Z - 18 T - 165 B**2 ), ( - 9 W + 15 T P + 20 S Z ), ( P W + 2 T Z - 11 B**3 ), ( 99 W - 11 B S + 3 B**2 ), ( 10000 B**2 + 6600 B + 2673 ) ) """;
# # 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);
# # jython examples for jas. # $Id$ # 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;
# 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;
import sys; from java.lang import System from jas import Ring, PolyRing, QQ, ZM, RF, GF from jas import terminate, startLog # polynomial examples: ideal prime decomposition in char p > 0, inseparable cases 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 - 2); #**2; f2 = (y**2 - c)**5; f3 = (z**2 - 2 * c); #**5; print "f1 = ", f1; print "f2 = ", f2;
# $Id$
#
import sys;
from jas import Ring, PolyRing, Ideal
from jas import QQ, ZZ
from jas import startLog, terminate
# example: Algebraic Statistics
# Drton, Sturmfels, Sullivant, example 2.1.3
#r = PolyRing(QQ(),"l1,l2",PolyRing.grad);
r = PolyRing(QQ(),"l1,l2",PolyRing.lex);
print "Ring: " + str(r);
print;
print one,l1,l2;
u0 = 3; u1 = 5; u2 = 7; u12 = 11;
f1 = (u1+u12)*(l1+l2+2)*(l1+1)*(l1+l2+1)\
+ (u12)*l1*(l1+1)*(l1+l2+1)\
- (u2+u12)*l1*(l1+l2+2)*(l1+l2+1)\
- (u0+u1+u2+u12)*l1*(l1+l2+2)*(l1+1) ;
f2 = (u2+u12)*(l1+l2+2)*(l2+1)*(l1+l2+1)\
+ (u12)*l2*(l2+1)*(l1+l2+1)\
- (u1+u12)*l2*(l1+l2+2)*(l1+l2+1)\
print "c^5:", c**5 + c.one()
print
c = CC((2, ), rn)
print "c: ", c
print "1/c: " + str(1 / c)
print
zm = ZM(19, 11)
print "zm: " + str(zm)
print "zm^2: " + str(zm * zm)
print "1/zm: " + str(1 / zm)
#print "zm.ring: " + str(zm.ring.toScript());
print
r = PolyRing(QQ(), "x,y", PolyRing.lex)
print "Ring: " + str(r)
print
# sage like: with generators for the polynomial ring
#is automatic: [one,x,y] = r.gens();
zero = r.zero()
try:
f = RF(r)
except:
f = None
print "f: " + str(f)
d = x**2 + 5 * x - 6
f = RF(r, d)
# # 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.
# $Id$
#
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
## 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);
# # jython examples for jas. # $Id$ # from java.lang import System 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;
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. # $Id$ # import sys; from jas import Ring, PolyRing, Ideal from jas import QQ, ZZ from jas import startLog, terminate # characteristic set example Circle of Apollonius, from CLO IVA r = PolyRing( QQ(), "u1,u2,x1,x2,x3,x4,x5,x6,x7,x8", PolyRing.lex ); print "Ring: " + str(r); print; #[one,u1,u2,x1,x2,x3,x4,x5,x6,x7,x8] = r.gens(); h1 = 2 * x1 - u1; h2 = 2 * x2 - u2; h3 = 2 * x3 - u1; h4 = 2 * x4 - u2; h5 = u2 * x5 + u1 * x6 - u1 * u2; h6 = u1 * x5 - u2 * x6; h7 = x1**2 - x2**2 - 2 * x1 * x7 + 2 * x2 * x8; h8 = x1**2 - 2 * x1 * x7 - x3**2 + 2 * x3 * x7 - x4**2 + 2 * x4 * x8; g = ( ( x5 - x7 )**2 + ( x6 - x8 )**2 - ( x1 - x7 )**2 - x8**2 ); L = [h1,h2,h3,h4,h5,h6,h7,h8];
# jython examples for jas. # $Id$ # from java.lang import System from jas import Ring, PolyRing, QQ, ZZ, GF from jas import terminate, startLog # polynomial examples: gcd #r = Ring( "Mod 1152921504606846883 (x,y,z) L" ); #r = Ring( "Rat(x,y,z) L" ); #r = Ring( "C(x,y,z) L" ); #r = Ring( "Z(x,y,z) L" ); r = PolyRing( QQ(), "x,y,z", PolyRing.lex ); print "Ring: " + str(r); print; #[one,x,y,z] = r.gens(); a = r.random(); b = r.random(); c = abs(r.random()); #c = 1; #a = 0; print "a = ", a; print "b = ", b; print "c = ", c;
# $Id$ # import sys; 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);
# # jython examples for jas. # $Id$ # from java.lang import System from jas import Ring, PolyRing, QQ, ZZ, GF, CC, ZM from jas import terminate, startLog # polynomial examples: Chines remainder theorem for polynomials #r = PolyRing( ZM(1152921504606846883), "(x,y,z)", PolyRing.lex ); #r = PolyRing( CC(), "(x,y,z)", PolyRing.lex ); ##r = PolyRing( ZZ(), "(x,y,z)", PolyRing.lex ); r = PolyRing(QQ(), "x,y,z", PolyRing.lex) print "Ring: " + str(r) print #[one,x,y,z] = r.gens(); a = r.random(3, 4) b = r.random(3, 3) c = abs(r.random(3, 3)) print "a = ", a print "b = ", b print "c = ", c print ff = [[a, b, c], [a + 1, b, c], [a, b + 1, c], [a, b, c + 1]]
import sys; 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 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;
# # 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 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;
# # 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;
#
# jython examples for jas.
# $Id$
#
from java.lang import System
from jas import Ring, PolyRing, QQ, DD
from jas import terminate, startLog
# polynomial examples: real roots over Q for zero dimensional ideals
r = PolyRing(QQ(),"q,w,s,x",PolyRing.lex);
print "Ring: " + str(r);
print;
#automatic: [one,q,w,s,x] = r.gens();
# EF(QQ()).realExtend("q","q^3 - 3", "[1,2]").realExtend("w", "w^2 - q", "[1,2]").realExtend("s", "s^5 - 2", "[1,2]").polynomial("x").build();
f1 = q**3 - 3;
f2 = w**2 - q;
#f3 = s**5 - 2;
f3 = s**3 - 2;
f4 = x**2 - w * s;
print "f1 = ", f1;
print "f2 = ", f2;
print "f3 = ", f3;
print "f4 = ", f4;
from jas import PolyRing, ZZ, QQ, ZM from jas import terminate, startLog 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];
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, 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;