m should be a monomial map between rings created by buildERing. Such a map can be constructed with buildEMonomialMap but this is not required.
For a map to ring R from ring S, the algorithm infers the entire equivariant map from where m sends the variable orbit generators of S. In particular for each orbit of variables of the form x(i1,...,ik), the image of x(0,...,k-1) is used.
egbToric uses an incremental strategy, computing Gröbner bases for truncations using FourTiTwo. Because of FourTiTwo’s efficiency, this strategy tends to be much faster than general equivariant Gröbner basis algorithms such as egb.
In the following example we compute an equivariant Gröbner basis for the vanishing equations of the second Veronese of Pn, i.e. the variety of n x n rank 1 symmetric matrices.
i1 : R = buildERing({symbol x}, {1}, QQ, 2); |
i2 : S = buildERing({symbol y}, {2}, QQ, 2); |
i3 : m = buildEMonomialMap(R,S,{x_0*x_1}) 2 2 o3 = map(R,S,{x , x x , x x , x }) 1 1 0 1 0 0 o3 : RingMap R <--- S |
i4 : G = egbToric(m, OutFile=>stdio) 3 -- used .00154826 seconds -- used .000178903 seconds (9, 9) new stuff found 4 -- used .00399308 seconds -- used .00165578 seconds (16, 26) new stuff found 5 -- used .00812619 seconds -- used .00513116 seconds (25, 60) 6 -- used .0207454 seconds -- used .0160127 seconds (36, 120) 7 -- used .0289227 seconds -- used .033651 seconds (49, 217) 2 o4 = {- y + y , - y y + y , - y y + y y , - y y + 1,0 0,1 1,1 0,0 1,0 2,1 0,0 2,0 1,0 2,1 1,0 ------------------------------------------------------------------------ y y , - y y + y y , - y y + y y , - y y + 2,0 1,1 2,2 1,0 2,1 2,0 3,2 1,0 3,0 2,1 3,2 1,0 ------------------------------------------------------------------------ y y } 3,1 2,0 o4 : List |
It is not checked if m is equivariant. Only the images of the orbit generators of the source ring are examined and the rest of the map ignored.