I have this code:
var("L E t c")
D=(1-c)*(L+3*E)
Dt=D-t*E
Dt_E=Dt*E;
Dt_E=Dt_E.expand()
print(Dt_E.subs(E*L==1/2, E*E==-1/6, L*L==-1/2, E*L*c==c*1/2))
I get:
-E*L*c + 1/2*c + 1/6*t
I get the same if I drop the last equality. What do I have to do for ELc to be replaced by c/2?
One can have a better control such substitutions using polynomial machinery, reducing a polynomial modulo the ideal generated by desired substitutions:
K.<L, E, t, c> = QQ[]
D=(1-c)*(L+3*E)
Dt=D-t*E
Dt_E=Dt*E
J = ideal([E*L-1/2, E*E+1/6, L*L+1/2])
print( Dt_E.reduce(J) )
Thank you. That's quite clever. May I ask if these rings are commutative by default?
Actually, it doesn't work as intended. In input:
R.<L, E, t, c> = PolynomialRing(QQ)
D=(1-c)*(L+3*E)
Dt=D-t*E
Dt_E=Dt*E
J = R.ideal([E*L-1/2, E*E+1/6, L*L+1/2])
print(Dt_E)
print(J)
print( Dt_E.reduce(J) )
The output is
-E^2*t - L*E*c - 3*E^2*c + L*E + 3*E^2
Ideal (L*E - 1/2, E^2 + 1/6, L^2 + 1/2) of Multivariate Polynomial Ring in L, E, t, c over Rational Field
0
But the last entry shouldn't be a 0. It should have $-t/6$ summand, shouldn't it?
That's because your substitutions are inconsistent with each other: multiplying E*E==-1/6, L*L==-1/2, gives E*E*L*L==1/12; however multiplying E*L==1/2 by itself gives E*E*L*L==1/4. In terms of the ideals, they generate the ideal J coinciding with the whole R, and thus everything reduces to 0 modulo J.
Ohhh I see. Thank you. I think the issue is that I am (should be?) working on a graded ring where negatively graded parts are 0 and where multiplication should really be "pairing", i.e if deg(E)=1, then deg(<e,e>)=0, bringing me one degree less, rather than up. So deg(EELL)=deg(<<<e,e>,L>,L>)=-2 and thus EELL=0.
So I guess this is not a solution for my problem. I know this is possible mathematically because it is what intersection theory on a surface (where the E, L are curves) in algebraic geometry/topology is about and it is actually defined using dimension of quotient rings (different rings) but I was hoping to be able to get around without having to implement all this (which is why I was using subs).
Alternative : Sympy's subsis recursive :
sage: Dt_E._sympy_().subs({u.lhs()._sympy_():u.rhs()._sympy_() for u in (E*L==1/2, E*E==-1/6, L*L==-1/2, E*L*c==c*1/2)})._sage_()
1/6*t
HTH,
I tried this but it fails:
AttributeError: 'Add' object has no attribute '_sympy_'
This is what I import:
from sage.symbolic.integration.integral import definite_integral
from sage.symbolic.assumptions import GenericDeclaration
from sympy import symbols
from sympy import expand
Asked: 2024-06-27 15:11:03 +0200
Seen: 227 times
Last updated: Jun 28
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I answer myself. If I add - in front of the ELC substitution, it works. I suppose this happens because for some reason sage is thinking E and -E are different things? How can I work over a commutative ring or even a field?