On Feb 21, 10:31 am, Robert Bradshaw <[email protected]>
wrote:
> k[x,y], k[x,z], k[y,z] all coerce to k[x,y,z], but there some overhead
> due to the extra object creation (assuming one needed to coerce back
> into the larger ring). This, and creation of the new ring, is probably
> still pretty cheap.

The problem I see is that the variable names are not the only
parameters involved in creating a new polynomial ring. How about term
ordering, underlying implementation etc? At some point I bet there are
going to be polynomial ring k[x,y,z] with custom term orderings. If
you compute a resultant of two elements there with respect to z, what
kind of k[x,y] are you going to create to hold the answer?

This problem does not arise in the univariate case, because the
prospective parent of the resultant already exists (i.e., is the base
ring).

Therefore, I think the safer choice is to let resultants of
multivariate polynomials end up in the same parent again. If people
want it in another ring, they can create that before hand and coerce
it in (apparently cheaply).

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