+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ vars = R.gens()
+ F = R.fraction_field()
+
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ L_x = matrix(F, n, n, L_x_i_j)
+ # Compute an extra power in case the rank is equal to
+ # the dimension (otherwise, we would stop at x^(r-1)).
+ x_powers = [ (L_x**k)*self.one().to_vector()
+ for k in range(n+1) ]
+ A = matrix.column(F, x_powers[:n])
+ AE = A.extended_echelon_form()
+ E = AE[:,n:]
+ A_rref = AE[:,:n]
+ r = A_rref.rank()
+ b = x_powers[r]
+
+ # The theory says that only the first "r" coefficients are
+ # nonzero, and they actually live in the original polynomial
+ # ring and not the fraction field. We negate them because
+ # in the actual characteristic polynomial, they get moved
+ # to the other side where x^r lives.
+ return -A_rref.solve_right(E*b).change_ring(R)[:r]
+
+ @cached_method
+ def rank(self):
+ r"""