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# binomialIdeal -- Compute the ideal of a monomial algebra

## 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 .