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Feature #947

IsRadical for ideal?

Added by John Abbott over 7 years ago. Updated 3 months ago.

Status:
In Progress
Priority:
Normal
Assignee:
-
Category:
New Function
Target version:
CoCoALib-0.99880
Start date:
18 Oct 2016
Due date:
% Done:

10%

Estimated time:
Spent time:
1.10 h

Description

Werner and Mario ask whether it could make sense to have an IsRadical function for ideals; can this be determined significantly more cheaply (at least in some cases) than actually computing the radical.

Related issues

Related to CoCoALib - Feature #796: CoCoALib function for radical (or SqFree) of a polynomialClosed2015-11-05

Related to CoCoA-5 - Slug #948: radical is slow (compared to singular) on these examplesClosed2016-10-18

History

#1 Updated by John Abbott over 7 years ago

  • Related to Feature #796: CoCoALib function for radical (or SqFree) of a polynomial added

#2 Updated by John Abbott over 7 years ago

  • Related to Slug #948: radical is slow (compared to singular) on these examples added

#3 Updated by Anna Maria Bigatti over 7 years ago

There is now "IsRadical" for 0-dimensional ideals. (and it's in our paper in progress ;-)
I've just realized I forgot to write the manual.... I write it now.

#4 Updated by Anna Maria Bigatti over 7 years ago

Apart from 0-dim (already implemented), and monomial, how can you determine it?

#5 Updated by John Abbott over 7 years ago

  • Status changed from New to In Progress
  • % Done changed from 0 to 10

The crude idea was just to run the usual code for computing the radical, and if at some point we determine with certainty that it is (or is not) radical then the code simply returns the relevant boolean. Of course, in many cases there may be only a small saving compared to computing the radical itself, but in some cases there may be a significant saving....?

#6 Updated by John Abbott over 7 years ago

I wonder if a probabilistic approach could work. If the ideal is 0-dim, the speed is adequate. If not, perhaps we could assign random values to certain indets until it is 0-dim (or anyway low dim); then if the image is radical the original probably was too? Presumably several random assignments should be made before accepting the result?

Would this work? Would it be usefully faster? Is my assertion "probably was too" correct (or at least vaguely correct)?

#7 Updated by Anna Maria Bigatti over 7 years ago

John Abbott wrote:

I wonder if a probabilistic approach could work. If the ideal is 0-dim, the speed is adequate. If not, perhaps we could assign random values to certain indets until it is 0-dim (or anyway low dim); then if the image is radical the original probably was too? Presumably several random assignments should be made before accepting the result?

Would this work? Would it be usefully faster? Is my assertion "probably was too" correct (or at least vaguely correct)?

Next paper ;-)

#8 Updated by John Abbott over 7 years ago

I think there could be problems with removing indets by giving them values: if there is an embedded 0-dim component then this will (almost certainly) be lost when removing indets (unless we happen to pick values corresponding to the coordindates of the 0-dim component).

:-/ :-(

#9 Updated by John Abbott about 4 years ago

  • Target version changed from CoCoALib-1.0 to CoCoALib-0.99800

#10 Updated by John Abbott over 3 years ago

  • Target version changed from CoCoALib-0.99800 to CoCoALib-0.99850

#11 Updated by John Abbott 3 months ago

  • Target version changed from CoCoALib-0.99850 to CoCoALib-0.99880

I think my idea might not work in general.
Let I := ideal(x+y^2) then I is radical. But if I add an extra generator x then I+ideal(x) = ideal(x,y^2) which
is not radical.

Let J := intersection(ideal(x,y^2), ideal(x+y+1)). Then J is not radical, but J+ideal(y) = ideal(y, x^2+x) which is radical.

I fear this shows that idea does not work (at least not in its simplest form).

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