# 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, , 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 => , 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 => , 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); -- 0.000253461 seconds elapsed -- 0.00188968 seconds elapsed -- 0.000462883 seconds elapsed -- 0.000208632 seconds elapsed -- 0.00183926 seconds elapsed -- 0.000601124 seconds elapsed -- 0.000211208 seconds elapsed -- 0.000225415 seconds elapsed -- 0.000517521 seconds elapsed -- 0.00025169 seconds elapsed -- 0.00195354 seconds elapsed -- 0.000599756 seconds elapsed -- 0.000223464 seconds elapsed -- 0.00188072 seconds elapsed -- 0.000592757 seconds elapsed -- 0.000231288 seconds elapsed -- 0.002035 seconds elapsed -- 0.000603711 seconds elapsed -- 0.000245057 seconds elapsed -- 0.00197612 seconds elapsed -- 0.000563458 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 -- 0.000232756 seconds elapsed -- 0.0022341 seconds elapsed -- 0.000599512 seconds elapsed -- 0.000249506 seconds elapsed -- 0.00206584 seconds elapsed -- 0.000559967 seconds elapsed -- 0.000256729 seconds elapsed -- 0.001899 seconds elapsed -- 0.000551549 seconds elapsed -- 0.000224734 seconds elapsed -- 0.00186772 seconds elapsed -- 0.000552165 seconds elapsed -- 0.000253889 seconds elapsed -- 0.00185302 seconds elapsed -- 0.000593969 seconds elapsed -- 0.000222335 seconds elapsed -- 0.00196526 seconds elapsed -- 0.000621705 seconds elapsed -- 0.000263126 seconds elapsed -- 0.00226363 seconds elapsed -- 0.000594095 seconds elapsed -- 0.000219172 seconds elapsed -- 0.00202803 seconds elapsed -- 0.000597765 seconds elapsed -- 0.000231649 seconds elapsed -- 0.00188589 seconds elapsed -- 0.000551547 seconds elapsed -- 0.000224147 seconds elapsed -- 0.00188076 seconds elapsed -- 0.000583668 seconds elapsed -- 0.000229128 seconds elapsed -- 0.00179718 seconds elapsed -- 0.000577852 seconds elapsed -- 0.000229741 seconds elapsed -- 0.00199495 seconds elapsed -- 0.00059398 seconds elapsed -- 0.000250173 seconds elapsed -- 0.00282162 seconds elapsed -- 0.000918691 seconds elapsed -- 0.00024722 seconds elapsed -- 0.00274833 seconds elapsed -- 0.000903925 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.