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);
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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 .00183405 seconds
-- used .000312977 seconds
(9, 9)
new stuff found
4
-- used .00429323 seconds
-- used .00183384 seconds
(16, 26)
new stuff found
5
-- used .0105897 seconds
-- used .00666985 seconds
(25, 60)
6
-- used .0236366 seconds
-- used .0174192 seconds
(36, 120)
7
-- used .0505432 seconds
-- used .0612128 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
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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.