Euler -- The Euler Characteristic

Synopsis

Usage:

Euler I

Euler(X,J)

Euler csm
Inputs:
- I, an ideal, a multi-homogeneous ideal in a graded polynomial ring over a field defining a closed subscheme V of \PP^{n_1}x...x\PP^{n_m}
- J, an ideal, an ideal in the graded polynomial ring which is coordinate ring of the Normal Toric Variety X
- X, a normal toric variety, a normal toric variety which is the ambient space that we are working in
- csm, a ring element, the CSM class of some variety V
Optional inputs:
- CompMethod (missing documentation) => ..., default value ProjectiveDegree, ProjectiveDegree, applicable for all cases where the methods in the package may be used
- CompMethod (missing documentation) => ..., default value ProjectiveDegree, PnResidual, this algorithm may be used for subschemes of \PP^n only
- Method (missing documentation) => ..., default value InclusionExclusion, InclusionExclusion, applicable for all inputs
- Method (missing documentation) => ..., default value InclusionExclusion, DirectCompleteInt, this method may provide a performance improvement when the input is a complete intersection, if the input is not a complete intersection inclusion/exclusion it will return an error
- InputIsSmooth (missing documentation) => ..., default value false, this option has values true/false and tells the method whether to assume the input ideal defines a smooth scheme, and hence to call the method Chern instead for reduced run time, alternatively the Chern function can be used directly
- Output => ..., default value ChowRingElement, the type of output to return the default output is an integer
- Output => ..., default value ChowRingElement, HashForm, the type of output to return, HashForm returns a MutableHashTable containing the key "CSM" (the CSM class), and keys of the form \{0\},\{1\},\{2\},...,\{0,1\},\{0,2\} ....\{0,1,2\}... and so on which correspond to the indices of the possible subsets of the generators of the input ideal, for each set of indices the CSM class of the hypersurface given by the product of all polynomials in the corresponding set of generators is stored, there is no extra cost to using this option
- IndsOfSmooth (missing documentation) => ..., default value {}, this option may speed up the run time when using the DirectCompleteInt Method if the user knows additional information about the input ideal, see IndsOfSmooth
Outputs:
- a ring element, the Euler characteristic

Description

For a subscheme V of an applicable toric variety X, this command computes the Euler characteristic

i1 : kk=ZZ/32749;

i2 : R=kk[x_0..x_4]

o2 = R

o2 : PolynomialRing

i3 : I=ideal(random(1,R),random(2,R))

                                                              2             
o3 = ideal (107x  + 4376x  - 6316x  + 3187x  + 3783x , - 6053x  + 8570x x  +
                0        1        2        3        4         0        0 1  
     ------------------------------------------------------------------------
           2                               2                                 
     10359x  - 16090x x  - 8210x x  + 5071x  + 8444x x  - 8997x x  - 6949x x 
           1         0 2        1 2        2        0 3        1 3        2 3
     ------------------------------------------------------------------------
             2                                                        2
     - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
             3         0 4        1 4         2 4         3 4         4

o3 : Ideal of R

i4 : time Euler(I,InputIsSmooth=>true)
     -- used 0.038085 seconds

o4 = 4

i5 : time Euler I
     -- used 0.0976904 seconds

o5 = 4

i6 : EulerIHash=Euler(I,Output=>HashForm);

i7 : A=ring EulerIHash#"CSM"

o7 = A

o7 : QuotientRing

i8 : EulerIHash#{0,1}==CSM(A,ideal(I_0*I_1))

o8 = true

i9 : J=I+ideal(x_0*x_2-x_3*x_0)

                                                              2             
o9 = ideal (107x  + 4376x  - 6316x  + 3187x  + 3783x , - 6053x  + 8570x x  +
                0        1        2        3        4         0        0 1  
     ------------------------------------------------------------------------
           2                               2                                 
     10359x  - 16090x x  - 8210x x  + 5071x  + 8444x x  - 8997x x  - 6949x x 
           1         0 2        1 2        2        0 3        1 3        2 3
     ------------------------------------------------------------------------
             2                                                        2
     - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x , x x 
             3         0 4        1 4         2 4         3 4         4   0 2
     ------------------------------------------------------------------------
     - x x )
        0 3

o9 : Ideal of R

Note that the ideal J above is a complete intersection, thus we may change the method option which may speed computation in some cases. We may also note that the ideal generated by the first 2 generators of I defines a smooth scheme and input this information into the method. This may also improve computation speed.

i10 : time Euler(J,Method=>DirectCompleteInt)
     -- used 0.0610646 seconds

o10 = 2

i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
     -- used 0.0491587 seconds

o11 = 2

Now consider an example in \PP^2 \times \PP^2.

i12 : R=MultiProjCoordRing({2,2})

o12 = R

o12 : PolynomialRing

i13 : r=gens R

o13 = {x , x , x , x , x , x }
        0   1   2   3   4   5

o13 : List

i14 : K=ideal(r_0^2*r_3-r_4*r_1*r_2,r_2^2*r_5)

              2              2
o14 = ideal (x x  - x x x , x x )
              0 3    1 2 4   2 5

o14 : Ideal of R

i15 : EulerK=Euler(K)

o15 = 7

i16 : csmK= CSM(K)

        2 2     2         2    2            2
o16 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
        1 2     1 2     1 2    1     1 2    2

      ZZ[h ..h ]
          1   2
o16 : ----------
         3   3
       (h , h )
         1   2

i17 : EulerK==Euler(csmK)

o17 = true

In the case where the ambient space is a toric variety which is not a product of projective spaces we must load the NormalToricVarieties package and must also input the toric variety. If the toric variety is a product of projective space it is recommended to use the form above rather than inputting the toric variety for efficiency reasons.

i18 : needsPackage "NormalToricVarieties"

o18 = NormalToricVarieties

o18 : Package

i19 : Rho = {{1,0,0},{0,1,0},{0,0,1},{-1,-1,0},{0,0,-1}}

o19 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {-1, -1, 0}, {0, 0, -1}}

o19 : List

i20 : Sigma = {{0,1,2},{1,2,3},{0,2,3},{0,1,4},{1,3,4},{0,3,4}}

o20 = {{0, 1, 2}, {1, 2, 3}, {0, 2, 3}, {0, 1, 4}, {1, 3, 4}, {0, 3, 4}}

o20 : List

i21 : X = normalToricVariety(Rho,Sigma,CoefficientRing =>ZZ/32749)

o21 = X

o21 : NormalToricVariety

i22 : CheckToricVarietyValid(X)

o22 = true

i23 : R=ring(X)

o23 = R

o23 : PolynomialRing

i24 : I=ideal(R_0^4*R_1,R_0*R_3*R_4*R_2-R_2^2*R_0^2)

              4       2 2
o24 = ideal (x x , - x x  + x x x x )
              0 1     0 2    0 2 3 4

o24 : Ideal of R

i25 : csmI=CSM(X,I)

        2       2
o25 = 5x x  + 3x  + 4x x  + x
        3 4     3     3 4    3

                      ZZ[x ..x ]
                          0   4
o25 : -----------------------------------------
      (x x , x x x , x  - x , x  - x , x  - x )
        2 4   0 1 3   0    3   1    3   2    4

i26 : EulerI=Euler(X,I)

o26 = 5

i27 : Euler(csmI)==EulerI

o27 = true

All the examples were done using symbolic computations with Gr\"obner bases. Changing the option CompMethod to bertini will do the main computations numerically, provided Bertini is installed and configured. Note that the bertini and PnResidual options may only be used for subschemes of \PP^n.

Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under probabilistic algorithm.

Ways to use Euler :

Euler(Ideal)
Euler(RingElement)

For the programmer

The object Euler is a method function with options.