egbToric -- computes the kernel of an equivariant monomial map

Synopsis

Usage:

G = egbToric m
Inputs:
- m, a ring map, an equivariant monomial map between rings with symmetric action
Optional inputs:
- OutFile => ..., default value null, where to send messages
Outputs:
- G, a list, an equivariant Gröbner basis for the kernel of the map

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 .00138383 seconds
     -- used 0 seconds
(9, 9)
new stuff found
4
     -- used .00291959 seconds
     -- used .00300074 seconds
(16, 26)
new stuff found
5
     -- used .00625445 seconds
     -- used .0200007 seconds
(25, 60)
6
     -- used .0136489 seconds
     -- used .141772 seconds
(36, 120)
7
     -- used .0282528 seconds
     -- used .620706 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.

Ways to use egbToric:

egbToric(RingMap)

For the programmer