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Design #634

Symbol in the coeff ring

Added by John Abbott over 9 years ago. Updated over 9 years ago.

Status:
Rejected
Priority:
Normal
Assignee:
John Abbott
Category:
enhancing/improving
Target version:
CoCoA-5.1.2 summer 2015
Start date:
22 Oct 2014
Due date:
% Done:

100%

Estimated time:
Spent time:
1.00 h

Description

Consider the following mini-session

QQab ::= QQ[a,b];
K := FractionField(QQab);
Use P ::= K[x,y];
a; // in which ring???

In which ring should the value of a lie?
Currently, it lies in the ring P so that expressions such as x+a are accepted (i.e. no problem of mixed rings).

If we allow automatic mapping from a subring to a superring then we could put the value of a in the smallest subring which contains it (in this case K), then in the expression x+a the value of a is automatically mapped into P before being added to x.

Related issues

Related to CoCoA-5 - Feature #7: Automatic mapping between (some) ringsResolved2011-10-20

Related to CoCoA-5 - Design #635: Automatic mapping of RingElem (in operation with a compound value)Closed2014-10-22

Related to CoCoA-5 - Design #636: Distinguish indets from symbols in coeffring in Use commandRejected2014-10-23

History

#1 Updated by John Abbott over 9 years ago

The problem was highlighted when Robbiano wanted to multiply a matrix (over K) by a; CoCoA-5 gave a mixed rings error. A simpler case would be a*LC(x+a); this will currently produce the same error, yet many would expect it to work.

#2 Updated by John Abbott over 9 years ago

There is a potential problem with my suggestion of symbols being mapped into the smallest containing subring. In the example I gave above, one would expect the expression 1/a to be evaluated without problem, but this won't happen if a is mapped into QQab.

So it would seem that I'd want the expression to be computed in the "smallest subring where the expression is valid" (or error if there is no such subring); this in turn would imply that the current ring would have to be passed as a parameter (so we know which subrings to consider)... I do not much like this. Consider continuing the session as follows:

x+1/a; // this *MUST* work
1/a; // this really should work too
a2 := a;
Use QQ[r,s];
1/a2; // what happens here?

If a2 lies in the smallest subring then its reciprocal cannot be computed, and it is not so obvious how to find the right superring to map it into.

Trickier than I thought...

#3 Updated by John Abbott over 9 years ago

  • Status changed from New to Rejected
  • Assignee set to John Abbott
  • % Done changed from 0 to 100
My previous comment convinces me that it is not feasible to make a have the value of the symbol a in the smallest subring. Furthermore it seems risky to me to change the current approach. For instance, one might want to perform the following:
deg(a);

How should CoCoA behave?
  1. it produces 0 (considering a as an element of P)
  2. it produces 1 (considering a as an element of QQab)
  3. it produces an error (considering a as an element of K)
    I believe that 3 is the least helpful. I can see arguments in favour of both 1 and 2, though my preference definitely lies with 1 because of the following scenario:
    a3 := (x+a)-x; // really just a
deg(a3); // surely gives 0
a3 = a;  // gives true, I hope!

    It would be quite a nasty surprise if the equality test gave any result other than true; and it would then be a nasty surprise if deg(a) = deg(a3) gave any result other than true.

I am now convinced that this idea should be rejected.

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