Macaulay2 » Documentation
Packages » MonomialAlgebras :: binomialIdeal
next | previous | forward | backward | up | index | toc

# binomialIdeal -- Compute the ideal of a monomial algebra

- Usage:
    `binomialIdeal P`
    `binomialIdeal B`
    `binomialIdeal M`
- Inputs:
  - P, a polynomial ring, multigraded, with B = degrees R and K = coefficientRing R, or
  - B, a list, In the case B is specified, K is set via the Option CoefficientField.
  - M, a MonomialAlgebra,
- Optional inputs:
  - CoefficientField => ..., default value ZZ/101, Option to set the coefficient field.
- Outputs:
  - an ideal,

## Description

Returns the toric ideal associated to the degree monoid B of the polynomial ring P as an ideal of P.

    i1 : kk=ZZ/101

    o1 = kk

    o1 : QuotientRing

    i2 : B = {{1,2},{3,0},{0,4},{0,5}}

    o2 = {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

    o2 : List

    i3 : S = kk[x_0..x_3, Degrees=> B]

    o3 = S

    o3 : PolynomialRing

    i4 : binomialIdeal S

                 3        2   6    2 3     4    3 2   5    4
    o4 = ideal (x x  - x x , x  - x x , x x  - x x , x  - x )
                 0 2    1 3   0    1 2   1 2    0 3   2    3

    o4 : Ideal of S

    i5 : C = {{1,2},{0,5}}

    o5 = {{1, 2}, {0, 5}}

    o5 : List

    i6 : P = kk[y_0,y_1, Degrees=> C]

    o6 = P

    o6 : PolynomialRing

    i7 : binomialIdeal P

    o7 = ideal ()

    o7 : Ideal of P

    i8 : M = monomialAlgebra B

    o8 = kk[x ..x ]
             0   3

    o8 : MonomialAlgebra generated by {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

    i9 : binomialIdeal M

                 3        2   6    2 3     4    3 2   5    4
    o9 = ideal (x x  - x x , x  - x x , x x  - x x , x  - x )
                 0 2    1 3   0    1 2   1 2    0 3   2    3

    o9 : Ideal of kk[x ..x ]
                      0   3

## Ways to use binomialIdeal:

- `binomialIdeal(List)`
- `binomialIdeal(MonomialAlgebra)`
- `binomialIdeal(PolynomialRing)`

## For the programmer

The object binomialIdeal is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MonomialAlgebras.m2:1394:0.