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_{(i_1,...,i_k)}, 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 P^n, i.e. the variety of n x n rank 1 symmetric matrices.
i1 : R = buildERing({symbol x}, {1}, QQ, 2);
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i2 : S = buildERing({symbol y}, {2}, QQ, 2);
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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
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i4 : G = egbToric(m, OutFile=>stdio)
3
-- used .00209799 seconds
-- used .000264356 seconds
(9, 9)
new stuff found
4
-- used .00473031 seconds
-- used .00215407 seconds
(16, 26)
new stuff found
5
-- used .0117192 seconds
-- used .00779032 seconds
(25, 60)
6
-- used .0230556 seconds
-- used .0181055 seconds
(36, 120)
7
-- used .0521714 seconds
-- used .0563704 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
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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.
The object egbToric is a method function with options.