poincare -- assemble degrees into polynomial

poincare -- assemble degrees into polynomial

poincare C -- encodes information about the degrees of basis elements of a free chain complex in a polynomial.
poincare M -- the same information about the free resolution of a module M.

The polynomial has a term (-1)^i T_0^(d_0) ... T_(n-1)^(d_(n-1)) in it for each basis element of C_i with multi-degree {d_0,...,d_(n-1)}. When the multi-degree has a single component, the term is (-1)^i T^(d_0).

The variable T is defined in a hidden local scope, so will print out as $T and not be directly accessible.

Note: the monomial ordering used in the degrees ring is RevLex, so the polynomials in it will be displayed with the smallest exponents first.

i1 : R = ZZ/101[x_0 .. x_3,y_0 .. y_3]



o1 = R



o1 : PolynomialRing

i2 : m = matrix table (2, 2, (i,j) -> x_(i+2*j))



o2 = | x_0 x_2 |

     | x_1 x_3 |



             2       2

o2 : Matrix R  <--- R

i3 : n = matrix table (2, 2, (i,j) -> y_(i+2*j))



o3 = | y_0 y_2 |

     | y_1 y_3 |



             2       2

o3 : Matrix R  <--- R

i4 : f = flatten (m*n - n*m)



o4 = | x_2y_1-x_1y_2 x_1y_0-x_0y_1+x_3y_1-x_1y_3 -x_2y_0+x_0y_2-x_3y_2+x_2y_3 -x_2y_1+x_1y_2 |



             1       4

o4 : Matrix R  <--- R

i5 : poincare cokernel f



            2      3

o5 = 1 - 3$T  + 2$T



o5 : ZZ[ZZ^1]

(cokernel f).poincare = p -- inform the system that the Poincare polynomial of the cokernel of f is p. This can speed the computation of a Groebner basis of f. For details, see computing Groebner bases.

See also:

degreesRing -- the ring of degrees

Ways to use poincare :

poincare ChainComplex

poincare Ideal

poincare Module

poincare MonomialIdeal

poincare Ring