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 .0027893 seconds
-- used .000452163 seconds
(9, 9)
new stuff found
4
-- used .00635838 seconds
-- used .00221455 seconds
(16, 26)
new stuff found
5
-- used .0131608 seconds
-- used .00806741 seconds
(25, 60)
6
-- used .0315382 seconds
-- used .0236558 seconds
(36, 120)
7
-- used .0585946 seconds
-- used .0980119 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.