sagbi -- Subalgebra basis (sagbi basis)

Synopsis

Usage:

N = sagbi M

N = sagbi A

N = sagbi L

N = sagbi B
Inputs:
- A, an instance of the type Subring,
- M, a matrix, of generators for a subring of a polynomial ring
- L, a list, containing generators for a subring of a polynomial ring
- B, an instance of the type SAGBIBasis, containing a partial computation of a sagbi basis
Optional inputs:
- Limit => an integer, default value 100, 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)
- Autosubduce => a Boolean value, default value true, flag indicating when to perform autosubduction on the generators before performing the Sagbi basis computation (See: Autosubduce)
- storePending => a Boolean value, default value true
Outputs:
- N, an instance of the type SAGBIBasis, a computation object holding the state of the sagbi basis computation

Description

The output of this function is a partial subalgebra basis stored in a computation object.

i1 : R = QQ[t_(1,1)..t_(3,3),MonomialOrder=>Lex];

i2 : M = genericMatrix(R,3,3);

             3       3
o2 : Matrix R  <--- R

i3 : A = subring gens minors(2, M);

i4 : (verifySagbi A)#"isSAGBI"

o4 = false

i5 : S = sagbi A;

i6 : gS = gens S

o6 = | t_(2,2)t_(3,3)-t_(2,3)t_(3,2) t_(2,1)t_(3,3)-t_(2,3)t_(3,1)
     ------------------------------------------------------------------------
     t_(2,1)t_(3,2)-t_(2,2)t_(3,1) t_(1,2)t_(3,3)-t_(1,3)t_(3,2)
     ------------------------------------------------------------------------
     t_(1,2)t_(2,3)-t_(1,3)t_(2,2) t_(1,1)t_(3,3)-t_(1,3)t_(3,1)
     ------------------------------------------------------------------------
     t_(1,1)t_(3,2)-t_(1,2)t_(3,1) t_(1,1)t_(2,3)-t_(1,3)t_(2,1)
     ------------------------------------------------------------------------
     t_(1,1)t_(2,2)-t_(1,2)t_(2,1)
     ------------------------------------------------------------------------
     t_(1,1)t_(2,2)t_(3,1)t_(3,3)-t_(1,1)t_(2,3)t_(3,1)t_(3,2)-t_(1,2)t_(2,1)
     ------------------------------------------------------------------------
     t_(3,1)t_(3,3)+t_(1,2)t_(2,3)t_(3,1)^2+t_(1,3)t_(2,1)t_(3,1)t_(3,2)-t_(1
     ------------------------------------------------------------------------
     ,3)t_(2,2)t_(3,1)^2 t_(1,1)t_(1,3)t_(2,2)t_(3,3)-t_(1,1)t_(1,3)t_(2,3)t
     ------------------------------------------------------------------------
     _(3,2)-t_(1,2)t_(1,3)t_(2,1)t_(3,3)+t_(1,2)t_(1,3)t_(2,3)t_(3,1)+t_(1,3
     ------------------------------------------------------------------------
     )^2t_(2,1)t_(3,2)-t_(1,3)^2t_(2,2)t_(3,1) |

             1       11
o6 : Matrix R  <--- R

i7 : (verifySagbi gS)#"isSAGBI"

o7 = true

Partial subalgebra bases are unavoidable since a subalgebra of a polynomial ring, endowed with some polynomial order, need not have a finite subalgebra basis. Here is a quintessential example of this phenomenon:

i8 : R=QQ[x,y];

i9 : A = subring matrix{{x+y,x*y,x*y^2}};

i10 : gens sagbi(A,Limit=>3)

o10 = | x+y xy xy2 |

              1       3
o10 : Matrix R  <--- R

i11 : gens sagbi(A,Limit=>10)

o11 = | x+y xy xy2 xy3 xy4 xy5 xy6 xy7 xy8 xy9 |

              1       10
o11 : Matrix R  <--- R

Nevertheless, a finite subalgebra basis can be computed in many cases.

Ways to use `sagbi` :

"sagbi(List)"
"sagbi(Matrix)"
"sagbi(SAGBIBasis)"
"sagbi(Subring)"

For the programmer

The object sagbi is a method function with options.

sagbi -- Subalgebra basis (sagbi basis)

Synopsis

Description

See also

Ways to use sagbi :

For the programmer

Ways to use `sagbi` :