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 | 1 | initial version |
Your equations can be turned into polynomial ones, and thus you'll will have better control over their solution if you empoy polynomial ideal machinery:
P.<r1, r2, r3, r4, s1, s2, s3, s4, s12, s13, s14, s23, s24, s34, w1, w2, w3, w4> = QQ[]
r = r1*w1 + r2*w2 + r3*w3 + r4*(1-w1-w2-w3)
v = ((w1**2*s1**2 + w2**2*s2**2 + w3**2*s3**2 + (1-w1-w2-w3)**2*s4**2+2*s12*w1*w2 + 2*s13*w1*w3+
2*s14*(1-w2-w1-w3)*w1+s23*w2*w3 + 2*s24*w2*(1-w2-w1-w3) + 2*s34*w3*(1-w2-w1-w3)))
dr1 = derivative(r, w1)
dr2 = derivative(r, w2)
dr3 = derivative(r, w3)
dv1 = derivative(v, w1)
dv2 = derivative(v, w2)
dv3 = derivative(v, w3)
eq1 = numerator(dv1/dr1 - 2*v/r)
eq2 = numerator(dv2/dr2 - 2*v/r)
eq3 = numerator(dv3/dr3 - 2*v/r)
eq4 = w1+w2+w3+w4 - 1
J = P.ideal( [eq1, eq2, eq3, eq4] )
I've used a variant of a custom myvariety() routine to obtain a list of 65 solutions (including parametric ones). You may find them at https://gist.github.com/maxale/76773bbae43ad953a4fef635c426d299
| | 2 | No.2 Revision |
Your equations can be turned into polynomial ones, and thus you'll will have better control over their solution if you empoy polynomial ideal machinery:
P.<r1, r2, r3, r4, s1, s2, s3, s4, s12, s13, s14, s23, s24, s34, w1, w2, w3, w4> = QQ[]
r = r1*w1 + r2*w2 + r3*w3 + r4*(1-w1-w2-w3)
v = ((w1**2*s1**2 + w2**2*s2**2 + w3**2*s3**2 + (1-w1-w2-w3)**2*s4**2+2*s12*w1*w2 + 2*s13*w1*w3+
2*s14*(1-w2-w1-w3)*w1+s23*w2*w3 + 2*s24*w2*(1-w2-w1-w3) + 2*s34*w3*(1-w2-w1-w3)))
dr1 = derivative(r, w1)
dr2 = derivative(r, w2)
dr3 = derivative(r, w3)
dv1 = derivative(v, w1)
dv2 = derivative(v, w2)
dv3 = derivative(v, w3)
eq1 = numerator(dv1/dr1 - 2*v/r)
eq2 = numerator(dv2/dr2 - 2*v/r)
eq3 = numerator(dv3/dr3 - 2*v/r)
eq4 = w1+w2+w3+w4 - 1
J = P.ideal( [eq1, eq2, eq3, eq4] )
I've used a variant of a custom myvariety() routine to obtain a list of 65 solutions (including parametric ones). ones) over the rationals. You may find them at https://gist.github.com/maxale/76773bbae43ad953a4fef635c426d299
| | 3 | No.3 Revision |
Your equations can be turned into polynomial ones, and thus you'll will have better control over their solution if you empoy polynomial ideal machinery:machinery - over the rationals it will look like:
P.<r1, r2, r3, r4, s1, s2, s3, s4, s12, s13, s14, s23, s24, s34, w1, w2, w3, w4> = QQ[]
r = r1*w1 + r2*w2 + r3*w3 + r4*(1-w1-w2-w3)
v = ((w1**2*s1**2 + w2**2*s2**2 + w3**2*s3**2 + (1-w1-w2-w3)**2*s4**2+2*s12*w1*w2 + 2*s13*w1*w3+ 2*s14*(1-w2-w1-w3)*w1+s23*w2*w3 + 2*s24*w2*(1-w2-w1-w3) + 2*s34*w3*(1-w2-w1-w3)))
dr1 = derivative(r, w1)
dr2 = derivative(r, w2)
dr3 = derivative(r, w3)
dv1 = derivative(v, w1)
dv2 = derivative(v, w2)
dv3 = derivative(v, w3)
eq1 = numerator(dv1/dr1 - 2*v/r)
eq2 = numerator(dv2/dr2 - 2*v/r)
eq3 = numerator(dv3/dr3 - 2*v/r)
eq4 = w1+w2+w3+w4 - 1
J = P.ideal( [eq1, eq2, eq3, eq4] )
I've used a variant of a custom myvariety() routine to obtain a list of 65 solutions (including parametric ones) over the rationals. You may find them at https://gist.github.com/maxale/76773bbae43ad953a4fef635c426d299
| | 4 | No.4 Revision |
Your equations can be turned into polynomial ones, and thus you'll will have better control over their solution if you empoy polynomial ideal machinery - over the rationals it will look like:
P.<r1, r2, r3, r4, s1, s2, s3, s4, s12, s13, s14, s23, s24, s34, w1, w2, w3, w4> = QQ[]
r = r1*w1 + r2*w2 + r3*w3 + r4*(1-w1-w2-w3)
v = ((w1**2*s1**2 + w2**2*s2**2 + w3**2*s3**2 + (1-w1-w2-w3)**2*s4**2+2*s12*w1*w2 + 2*s13*w1*w3+ 2*s14*(1-w2-w1-w3)*w1+s23*w2*w3 + 2*s24*w2*(1-w2-w1-w3) + 2*s34*w3*(1-w2-w1-w3)))
dr1 = derivative(r, w1)
dr2 = derivative(r, w2)
dr3 = derivative(r, w3)
dv1 = derivative(v, w1)
dv2 = derivative(v, w2)
dv3 = derivative(v, w3)
eq1 = numerator(dv1/dr1 - 2*v/r)
eq2 = numerator(dv2/dr2 - 2*v/r)
eq3 = numerator(dv3/dr3 - 2*v/r)
eq4 = w1+w2+w3+w4 - 1
J = P.ideal( [eq1, eq2, eq3, eq4] )
I've used a variant of a custom myvariety() routine to obtain a list of 65 solutions (including parametric ones) over the rationals. rationals, although it may be incomplete. You may find them at https://gist.github.com/maxale/76773bbae43ad953a4fef635c426d299
| | 5 | No.5 Revision |
Your equations can be turned into polynomial ones, and thus you'll will have better control over their solution if you empoy polynomial ideal machinery - over the rationals it will look like:
P.<r1, r2, r3, r4, s1, s2, s3, s4, s12, s13, s14, s23, s24, s34, w1, w2, w3, w4> = QQ[]
r = r1*w1 + r2*w2 + r3*w3 + r4*(1-w1-w2-w3)
v = ((w1**2*s1**2 + w2**2*s2**2 + w3**2*s3**2 + (1-w1-w2-w3)**2*s4**2+2*s12*w1*w2 + 2*s13*w1*w3+ 2*s14*(1-w2-w1-w3)*w1+s23*w2*w3 + 2*s24*w2*(1-w2-w1-w3) + 2*s34*w3*(1-w2-w1-w3)))
dr1 = derivative(r, w1)
dr2 = derivative(r, w2)
dr3 = derivative(r, w3)
dv1 = derivative(v, w1)
dv2 = derivative(v, w2)
dv3 = derivative(v, w3)
eq1 = numerator(dv1/dr1 - 2*v/r)
eq2 = numerator(dv2/dr2 - 2*v/r)
eq3 = numerator(dv3/dr3 - 2*v/r)
eq4 = w1+w2+w3+w4 - 1
J = P.ideal( [eq1, eq2, eq3, eq4] )
I've used a variant of a custom myvariety() routine to obtain a list of 65 solutions (including parametric ones) over the rationals, although it may be incomplete. You Among them there are 37 solutions with nonzero dr1, dr2, dr3, and r (appearing in denominators of original equations), which you may find them at https://gist.github.com/maxale/76773bbae43ad953a4fef635c426d299
