# 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 );
# 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; f1 = (x**2 + 2)**2 f2 = (y**2 - x)**5
# polynomial examples: ideal radical decomposition, example 8.16 in GB book, base field with p-th root # noThreads(); # must be called very early prime = 5 cf = GF(prime) #cf = QQ(); ca = PolyRing(cf, "t", PolyRing.lex) print "ca = " + str(ca) [ea, ta] = ca.gens() print "ea = " + str(ea) print "ta = " + str(ta) print Qpt = RF(ca) #print Qpt.gens(); [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
from jas import QQ, ZZ, GF, RF, startLog, terminate
startLog();
# solvable module example
#mod=17;
mod=32003;
#mod=536870909;
#mod=1152921504606846883;
coeff = GF(mod)
#coeff = QQ()
p = PolyRing(
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)),
c = CC( (2,),rn );
print "c:", c;
print;
r = Ring( "Q(x,y) L" );
print "Ring: " + str(r);
print;
# sage like: with generators for the polynomial ring
[x,y] = r.gens();
one = r.one();
zero = r.zero();
try:
f = RF();
except:
f = None;
print "f: " + str(f);
d = x**2 + 5 * x - 6;
f = RF(d);
print "f: " + str(f);
n = d*d + y + 1;
f = RF(d,n);
print "f: " + str(f);
print;
# beware not to mix expressions
f = f**2 - f;
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)
print "f: " + str(f)
n = d * d + y + 1
f = RF(r, d, n)
print "f: " + str(f)
print
# beware not to mix expressions
f = f**2 - f
#r = PolyRing(QQ(),"B,S,T,Z,P,W",PolyRing.lex); #r = PolyRing(CC(),"B,S,T,Z,P,W",PolyRing.lex); #r = PolyRing(DD(),"B,S,T,Z,P,W",PolyRing.lex); #r = PolyRing(ZM(19),"B,S,T,Z,P,W",PolyRing.lex); #r = PolyRing(ZM(1152921504606846883),"B,S,T,Z,P,W",PolyRing.lex); # 2^60-93 #rc = PolyRing(ZZ(),"e,f",PolyRing.lex); #rc = PolyRing(QQ(),"e,f",PolyRing.lex); #r = PolyRing(rc,"B,S,T,Z,P,W",PolyRing.lex); rqc = PolyRing(ZZ(), "e,f", PolyRing.lex) print "Q-Ring: " + str(rqc) print "rqc.gens() = ", [str(f) for f in rqc.gens()] print [pone, pe, pf] = rqc.gens() r = PolyRing(RF(rqc), "B,S,T,Z,P,W", PolyRing.lex) print "Ring: " + str(r) print # sage like: with generators for the polynomial ring print "r.gens() = ", [str(f) for f in r.gens()] print [one, e, f, B, S, T, Z, P, W] = r.gens() #[one,B,S,T,Z,P,W] = r.gens(); #[one,I,B,S,T,Z,P,W] = r.gens(); 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
## 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
c = CC( (2,),rn );
print "c:", c;
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);
print "f: " + str(f);
n = d*d + y + 1;
f = RF(r,d,n);
print "f: " + str(f);
print;
# beware not to mix expressions
f = f**2 - f;
fu = (u**2+u+1)**p;
print "fu = ", fu;
print;
t = System.currentTimeMillis();
G = cr.squarefreeFactors(fu);
t = System.currentTimeMillis() - t;
#print "G = ", G; #.toScript();
print "factor time =", t, "milliseconds";
for h, i in G.iteritems():
print "h**i = (", h, ")**" + str(i);
h = h**i;
print;
qcr = RF(cr);
print "Ring qcr: " + str(qcr.factory());
#not ok#r = PolyRing(cr,"x",PolyRing.lex );
r = PolyRing(qcr,"x",PolyRing.lex );
print "Ring r: " + str(r);
#qr = RF(r);
#print "Ring qr: " + str(qr.factory());
print;
#[one,u,x] = r.gens();
print "one = " + str(one);
print "u = " + str(u);
print "x = " + str(x);
print "s2 = " + str(s2)
s3 = z**2 + (2) * y * z + (2) * x * z + y**2 + (
2) * x * y + (b**2 + 2 * a * b + a**2 + (4, 5) * b + (4, 5) * a +
(4, 25)) * y + x**2 - (c - (2, 3)) * x + ((1, 2))
print "s3 = " + str(s3)
s4 = s1 + s2 - 2 * s3
print "s4 = " + str(s4)
print "s4.factory() = " + str(s4.factory())
x = PolyRing(PolyRing(QQ(), "a, b, c", PolyRing.grad), "x, y, z", PolyRing.lex)
print "x = " + str(x)
print
print "------- RF(PolyRing(ZZ(),\"a,b,c\",PolyRing.lex)) ---------"
r = PolyRing(ZZ(), "a,b,c", PolyRing.lex)
print "r = " + str(r)
rf = RF(r)
print "rf = " + str(rf.factory())
[one, a, b, c] = rf.gens()
print "one = " + str(one)
print "a = " + str(a)
print "b = " + str(b)
print "c = " + str(c)
q1 = a / b
print "q1 = " + str(q1)
q2 = (-2 * c**2 + 4 * b**2 + 4 * a**2 - 7)
print "q2 = " + str(q2)
q3 = (-7 * b + 4 * a + 12)
print "q3 = " + str(q3)
q4 = q2 / q3
print "q4 = " + str(q4)
q5 = (2 * c**2 - 4 * b**2 - 4 * a**2 + 7) / (7 * b - 4 * a - 12)
# # jython examples for jas. # $Id$ # from java.lang import System from jas import PolyRing, QQ, ZM, RF, GF from jas import terminate, startLog # polynomial examples: squarefree: characteristic p, infinite r = PolyRing(RF(PolyRing(GF(5), "u, v", PolyRing.lex)), "x y z", PolyRing.lex) print "r = " + str(r) #automatic: [one,u,v,x,y,z] = r.gens(); print "one = " + str(one) print "u = " + str(u) print "v = " + str(v) print "x = " + str(x) print "y = " + str(y) print "z = " + str(z) print "Ring: " + str(r) print a = r.random(k=1, l=3) b = r.random(k=1, l=3) c = r.random(k=1, l=3) if a.isZERO():
fu = (u**2 + u + 1)**p
print "fu = ", fu
print
t = System.currentTimeMillis()
G = cr.squarefreeFactors(fu)
t = System.currentTimeMillis() - t
#print "G = ", G; #.toScript();
print "factor time =", t, "milliseconds"
for h, i in G.iteritems():
print "h**i = (", h, ")**" + str(i)
h = h**i
print
qcr = RF(cr)
print "Ring qcr: " + str(qcr.factory())
#not ok#r = PolyRing(cr,"x",PolyRing.lex );
r = PolyRing(qcr, "x", PolyRing.lex)
print "Ring r: " + str(r)
#qr = RF(r);
#print "Ring qr: " + str(qr.factory());
print
#[one,u,x] = r.gens();
print "one = " + str(one)
print "u = " + str(u)
print "x = " + str(x)
s2 = (1,2) + ( (2,3) - c ) * x + ( (2,5) + a + b )**2 * y + ( x + y + z )**2; print "s2 = " + str(s2); s3 = z**2 + ( 2 ) * y * z + ( 2 ) * x * z + y**2 + ( 2 ) * x * y + ( b**2 + 2 * a * b + a**2 + (4,5) * b + (4,5) * a + (4,25) ) * y + x**2 - ( c - (2,3) ) * x + ( (1,2) ); print "s3 = " + str(s3); s4 = s1 + s2 - 2 * s3; print "s4 = " + str(s4); print "s4.factory() = " + str(s4.factory()); x = PolyRing(PolyRing(QQ(),"a, b, c",PolyRing.grad),"x, y, z",PolyRing.lex); print "x = " + str(x); print; print "------- RF(PolyRing(ZZ(),\"a,b,c\",PolyRing.lex)) ---------"; r = PolyRing(ZZ(),"a,b,c",PolyRing.lex); print "r = " + str(r); rf = RF(r); print "rf = " + str(rf.factory()); [one,a,b,c] = rf.gens(); print "one = " + str(one); print "a = " + str(a); print "b = " + str(b); print "c = " + str(c); q1 = a / b; print "q1 = " + str(q1); q2 = ( -2 * c**2 + 4 * b**2 + 4 * a**2 - 7 ); print "q2 = " + str(q2); q3 = ( -7 * b + 4 * a + 12 ); print "q3 = " + str(q3); q4 = q2 / q3; print "q4 = " + str(q4); q5 = ( 2 * c**2 - 4 * b**2 - 4 * a**2 + 7 ) / (7 * b - 4 * a - 12 );
# polynomial examples: ideal radical decomposition, example 8.16 in GB book, base field with p-th root # noThreads(); # must be called very early prime = 5; cf = GF(prime); #cf = QQ(); ca = PolyRing(cf,"t",PolyRing.lex); print "ca = " + str(ca); [ea,ta] = ca.gens(); print "ea = " + str(ea); print "ta = " + str(ta); print; Qpt = RF(ca); #print Qpt.gens(); [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;
s2 = (1,2) + ( (2,3) - c ) * x + ( (2,5) + a + b )**2 * y + ( x + y + z )**2; print "s2 = " + str(s2); s3 = z**2 + ( 2 ) * y * z + ( 2 ) * x * z + y**2 + ( 2 ) * x * y + ( b**2 + 2 * a * b + a**2 + (4,5) * b + (4,5) * a + (4,25) ) * y + x**2 - ( c - (2,3) ) * x + ( (1,2) ); print "s3 = " + str(s3); s4 = s1 + s2 - 2 * s3; print "s4 = " + str(s4); print "s4.factory() = " + str(s4.factory()); x = PolyRing(PolyRing(QQ(),"a, b, c",PolyRing.grad),"x, y, z",PolyRing.lex); print "x = " + str(x); print; print "------- RF(PolyRing(ZZ(),\"a,b,c\",PolyRing.lex)) ---------"; r = PolyRing(ZZ(),"a,b,c",PolyRing.lex); print "r = " + str(r); rf = RF(r); print "rf = " + str(rf.factory()); [one,a,b,c] = rf.gens(); print "one = " + str(one); print "a = " + str(a); print "b = " + str(b); print "c = " + str(c); q1 = a / b; print "q1 = " + str(q1); q2 = ( -2 * c**2 + 4 * b**2 + 4 * a**2 - 7 ); print "q2 = " + str(q2); q3 = ( -7 * b + 4 * a + 12 ); print "q3 = " + str(q3); q4 = q2 / q3; print "q4 = " + str(q4); q5 = ( 2 * c**2 - 4 * b**2 - 4 * a**2 + 7 ) / (7 * b - 4 * a - 12 );
c = CC((2, ), rn)
print "c:", c
print
r = Ring("Q(x,y) L")
print "Ring: " + str(r)
print
# sage like: with generators for the polynomial ring
[x, y] = r.gens()
one = r.one()
zero = r.zero()
try:
f = RF()
except:
f = None
print "f: " + str(f)
d = x**2 + 5 * x - 6
f = RF(d)
print "f: " + str(f)
n = d * d + y + 1
f = RF(d, n)
print "f: " + str(f)
print
# beware not to mix expressions
f = f**2 - 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
from java.lang import System
from jas import Ring, PolyRing
from jas import ZM, QQ, AN, RF
from jas import terminate, startLog
# polynomial examples: factorization over Z_p(x)(sqrt{p}(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()
#print "ewx = " + str(ewx);
print "ax = " + str(ax)
print "wx = " + str(wx)
print
#rootx = wx**5 - 2; # not working
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); print "Qr = " + str(Qr.factory()); [er,wr,ar] = Qr.gens(); #print "er = " + str(er); #print "wr = " + str(wr); #print "ar = " + str(ar); print; Qwx = PolyRing(Qr,"wx",PolyRing.lex); print "Qwx = " + str(Qwx); [ewx,wwx,ax,wx] = Qwx.gens(); #print "ewx = " + str(ewx); print "ax = " + str(ax); #print "wwx = " + str(wwx); print "wx = " + str(wx); print;
import sys from java.lang import System from jas import PolyRing, QQ, RF, DD from jas import terminate, startLog # system biology examples: GB in RF() # see: Informatik Spektrum, 2009, February, # Laubenbacher, Sturmfels: Computer Algebra in der Systembiologie # example from: http://theory.bio.uu.nl/rdb/books/tb.pdf # ------------ input rational expression -------------------- r = PolyRing(RF(PolyRing(QQ(), "A", PolyRing.lex)), "L, M, R", PolyRing.lex) print "PolyRing: " + str(r) print #automatic: [one,A,L,M,R] = r.gens(); c = 1 gamma = 1 v = 1 c0 = QQ(5, 100) # 0.05 h = 2 n = 5 delta = QQ(2, 10) # 0.2
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(); #print "ewx = " + str(ewx); print "ax = " + str(ax); print "wx = " + str(wx); print; #rootx = wx**5 - 2; # not working
# polynomial examples: ideal prime decomposition
# TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
# Volume 296, Number 2. August 1986
# ON THE DEPTH OF THE SYMMETRIC ALGEBRA
# J. HERZOG M. E. ROSSI AND G. VALLA
#r = PolyRing(QQ(),"x,t,z,y",PolyRing.lex);
#r = PolyRing(QQ(),"x,y,t,z",PolyRing.lex);
#r = EF(QQ()).extend("x").polynomial("y,z,t").build(); #,PolyRing.lex);
#c = PolyRing(QQ(),"t,y",PolyRing.lex);
#c = PolyRing(ZM(32003,0,True),"t",PolyRing.lex);
#c = PolyRing.new(GF(32003),"t",PolyRing.lex);
c = PolyRing(ZZ(), "t", PolyRing.lex)
r = PolyRing(RF(c), "z,y,x", PolyRing.lex)
print "Ring: " + str(r)
print
#automatic: [one,t,z,y,x] = r.gens();
print "one = ", one
print "x = ", x
print "y = ", y
print "z = ", z
print "t = ", t
f1 = x**3 - y**7
f2 = x**2 * y - x * t**3 - z**6
f3 = z**2 - t**3
#f3 = z**19 - t**23;