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 .00129934 seconds -- used .000153938 seconds (9, 9) new stuff found 4 -- used .00228987 seconds -- used .000998833 seconds (16, 26) new stuff found 5 -- used .00538654 seconds -- used .00362318 seconds (25, 60) 6 -- used .0116682 seconds -- used .00958138 seconds (36, 120) 7 -- used .0263626 seconds -- used .031241 seconds (49, 217) 2 o4 = {- y + y , - y y + y , - y y + y 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 2,0 1,1 2,2 1,0 2,1 2,0 3,2 1,0 ---------------------------------------------------------------------------------------------------------------------------- y y , - y y + y y } 3,0 2,1 3,2 1,0 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.