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egbToric -- computes the kernel of an equivariant monomial map

Synopsis

Description

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);
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 .000930124 seconds
     -- used 0 seconds
(9, 9)
new stuff found
4
     -- used .00291486 seconds
     -- used .0033292 seconds
(16, 26)
new stuff found
5
     -- used .00556498 seconds
     -- used .0196546 seconds
(25, 60)
6
     -- used .0131031 seconds
     -- used .180352 seconds
(36, 120)
7
     -- used .0271448 seconds
     -- used .668753 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

Caveat

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.

See also

Ways to use egbToric :

For the programmer

The object egbToric is a method function with options.