## Feature #1030

### IsInRadical: case of homog ideal

**Description**

Currently `IsInRadical`

is defined in a CPKG5, but may soon be translated to CoCoALib.

The case of a homog poly in a homog ideal is handled specially.

I wonder if we cannot improve it by simply testing each homog component of the poly for membership in the radical.

**Related issues**

### History

#### #1 Updated by John Abbott about 1 year ago

**Subject**changed from*IsInRadical: case of homog poly*to*IsInRadical: case of homog ideal***Status**changed from*New*to*In Progress***% Done**changed from*0*to*10*

I think that if the ideal `I`

is homog then `IsInRadical(f,I)`

is the same as the logical-and of `IsInRadical(f_d,I)`

for all homog components `f_d`

of `f`

. I do not know whether it is faster to compute it this way... or maybe my maths is wrong?

#### #2 Updated by Anna Maria Bigatti about 1 year ago

John Abbott wrote:

I think that if the ideal

`I`

is homog then`IsInRadical(f,I)`

is the same as the logical-and of`IsInRadical(f_d,I)`

for all homog components`f_d`

of`f`

. I do not know whether it is faster to compute it this way... or maybe my maths is wrong?

correct: f = f_d + .... (f_d homog of deg d = deg(f)).

f^n = (f_d)^n + ... in I implies (f_d)^n in I implies f_d is in radical(I)

#### #3 Updated by John Abbott about 1 year ago

**Related to***Bug #1032: IsInRadical: fragile code*added

#### #4 Updated by John Abbott about 1 year ago

**Related to***Feature #1033: Split poly into homog parts*added

#### #5 Updated by John Abbott 9 months ago

**Status**changed from*In Progress*to*Feedback***Assignee**set to*John Abbott***Target version**changed from*CoCoALib-0.99999*to*CoCoALib-0.99560***% Done**changed from*10*to*90*

The CoCoA-5 package was translated into C++ by some students at Kassel.

I have cleaned up the resulting code, and checked it in: see files `RadicalMembership`

I have added doc and a test (but no example).

I have made the fns available via CoCoA-5; the old package is still there, but I have changed the fn names to avoid clashes. Probably the package should simply be deleted (perhaps after a bit more testing?)

I have added a couple of "heuristic tricks" to ** IsInRadical**, as otherwise computation times can be very long (esp, when the polynomial is not in the radical). The trick is just to see if a generator (or RGB element) is not square-free; if so, add as new generator the "radical" of that generator.

Note that `SqFreeFactor`

can be slow when coeffs are in a finite field (since GCD is still via a GBasis computation); so the trick is not applied to "big" polys.

#### #6 Updated by John Abbott 9 months ago

- if the ideal is not 0-dim, adjoin some random linear polys (or linear forms) to the gens possibly making the ideal 0-dim, then test for radical membership. If poly is not in radical of extended ideal, it is surely not in the radical of the original ideal; the other case is less clear.
- if ideal is over QQ, try a modular approach; perhaps use
`MinPowerInIdeal`

to predict power of original power which ought to be in the original ideal (and then test that power directly).

Is it true that every (polynomial) ideal *I* has an "exponent" *exp(I)* such that for any polynomial `IsInRadical(f,I)`

iff `f^exp(I) IsIn I`

. I'm not sure how the exponent could be computed.

#### #7 Updated by John Abbott 5 months ago

**Status**changed from*Feedback*to*Closed***% Done**changed from*90*to*100*

#### #8 Updated by John Abbott 5 months ago

**Description**updated (diff)