> However, I am not sure if it is efficient to use these commands often.
> Suppose f is a map from basering to some other ring R, and I, J are
> ideals in R. Then, preimage(R,f,I) computes the preimage of I under f.
> I guess internally some Groebner basis computation is done. When you
> then do preimage(R,f,J), I wonder if the same Groebner basis
> computation is repeated, or if the first result was cached.
A priori, the two computations (one with I, one with J) are different.
Computing a kernel is really nothing more than computing a groebner
basis for a well-chosen ideal, which depends on I or J. It's not
inconceivable that some clever person could cache some intermediate
results, but it's not entirely straightforward.
Also, with groebner basis being available in sage, it's easy to
implement at least a naive version in pure sage which should be just
as fast as any other. Here's a crap one (don't laugh):
def kernel(phi, R, J, S, I):
"""phi: list of elements of polyring S, defining a homomorphism from
polyring R to S by specifying the images of the generators.
This computes the kernel of the induced map R/J --> S/I.
Limitations: variables in R and S must have different names. Silly,
but it'll do."""
Y= R.gens()
X= S.gens()
k= R.base_ring()
A= PolynomialRing(k, X + Y, order= "lex")
L= [A(P) for P in I.gens()] + [ A(Y[i]) - A(phi[i]) for i in range(len
(phi))]
K= L*A
ans= K.groebner_basis()
Yvars= A.gens()[len(X):]
def only_Y_vars(P):
for var in P.variables():
if var not in Yvars:
return False
return True
ans= filter(only_Y_vars, ans)
return [R(P) for P in ans]
Given this one ought to :
(i) write a proper version which has no silly name limitation, uses
term orders cleverly, uses elimination_ideal, etc : all relatively
straightforward.
(ii) add the functionality to SAGE : a lot of trouble as far as i'm
concerned.
Hey if anyone wants to take me through the steps for (ii)... you
wouldn't believe how little i know.
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