subringIntersection -- Intersection of subrings

Synopsis

Usage:

result = subringIntersection(S1, S2)
Inputs:
- S1, an instance of the type Subring,
- S2, an instance of the type Subring,
- SAGBILimitType, a string, Either \"Fixed\" or \"Function\". Determined the stopping criterion for the sagbi computation. If \"Fixed\" then the Limit is used, otherwise if \"Function\" is selected then the maximum degree of the input generators is used as the degree limit.
Optional inputs:
- Strategy => a string, default value "Master", the update strategy at the beginning of each loop: \"DegreeByDegree\", \"Incremental\", and \"Master\". The strategy \"Master\" is a hybrid that combines the other two; starting with \"DegreeByDegree\" for low degrees and switching to \"Incremental\". (See: Strategy)
- SubductionMethod => a string, default value "Top", the method used for subduction either: \"Top\" or \"Engine\". (See: SubductionMethod)
- Limit => an integer, default value 20, a degree limit for the binomial S-pairs that are computed internally.
- PrintLevel => an integer, default value 0, When this is greater than zero, information is printed about the progress of the computation (See: PrintLevel).
- SAGBILimitType
Outputs:
- result, an instance of the type Subring,

Description

Computes the intersection of subrings "S_1" and "S_2". These subrings must be subrings of the same ambient ring. The ambient ring is allowed to be a polynomial ring or the quotient of a polynomial ring.

i1 : R = QQ[x,y];

i2 : I = ideal(x^3 + x*y^2 + y^3);

o2 : Ideal of R

i3 : Q = R/I;

i4 : S1 = subring {x^2, x*y};

i5 : S2 = subring {x, y^2};

i6 : S = subringIntersection(S1, S2);
 -- 0.000092062 seconds elapsed
 -- 0.00118691 seconds elapsed
 -- 0.000287496 seconds elapsed
 -- 0.000095297 seconds elapsed
 -- 0.00102847 seconds elapsed
 -- 0.000256859 seconds elapsed
 -- 0.000096009 seconds elapsed
 -- 0.000082133 seconds elapsed
 -- 0.000212487 seconds elapsed
 -- 0.000097542 seconds elapsed
 -- 0.000951513 seconds elapsed
 -- 0.000240097 seconds elapsed
 -- 0.000095649 seconds elapsed
 -- 0.000880971 seconds elapsed
 -- 0.000244706 seconds elapsed
 -- 0.000095919 seconds elapsed
 -- 0.000888896 seconds elapsed
 -- 0.000228716 seconds elapsed
 -- 0.000092343 seconds elapsed
 -- 0.000988472 seconds elapsed
 -- 0.000232263 seconds elapsed
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction

i7 : gens S

o7 = | x2 x2y2+xy3 y4 xy3 y6 xy5 |

             1       6
o7 : Matrix Q  <--- Q

i8 : isSAGBI S
 -- 0.000113041 seconds elapsed
 -- 0.00107184 seconds elapsed
 -- 0.000228777 seconds elapsed
 -- 0.000088676 seconds elapsed
 -- 0.000977671 seconds elapsed
 -- 0.000229207 seconds elapsed
 -- 0.00013391 seconds elapsed
 -- 0.000953758 seconds elapsed
 -- 0.000270504 seconds elapsed
 -- 0.000089517 seconds elapsed
 -- 0.00088026 seconds elapsed
 -- 0.00022562 seconds elapsed
 -- 0.000092342 seconds elapsed
 -- 0.000867606 seconds elapsed
 -- 0.000216724 seconds elapsed
 -- 0.000088566 seconds elapsed
 -- 0.000903263 seconds elapsed
 -- 0.000222755 seconds elapsed
 -- 0.000089798 seconds elapsed
 -- 0.00108306 seconds elapsed
 -- 0.000224929 seconds elapsed
 -- 0.000090168 seconds elapsed
 -- 0.000956412 seconds elapsed
 -- 0.000223326 seconds elapsed
 -- 0.000088816 seconds elapsed
 -- 0.000869199 seconds elapsed
 -- 0.000225631 seconds elapsed
 -- 0.000089547 seconds elapsed
 -- 0.000868498 seconds elapsed
 -- 0.000220159 seconds elapsed
 -- 0.000086982 seconds elapsed
 -- 0.000852307 seconds elapsed
 -- 0.000229467 seconds elapsed
 -- 0.000089717 seconds elapsed
 -- 0.000896902 seconds elapsed
 -- 0.000217737 seconds elapsed
 -- 0.000106708 seconds elapsed
 -- 0.0013266 seconds elapsed
 -- 0.000361193 seconds elapsed
 -- 0.000091511 seconds elapsed
 -- 0.00139086 seconds elapsed
 -- 0.000395937 seconds elapsed
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction
timing raw subduction

o8 = true

If the generators of $S$ form a sagbi basis and the degree limit is high enough, then they are a generating set for the intersection.

Ways to use subringIntersection :

"subringIntersection(Subring,Subring)"

For the programmer

The object subringIntersection is a method function with options.

subringIntersection -- Intersection of subrings

Synopsis

Description

See also

Ways to use subringIntersection :

For the programmer