From d65df87de905c6ba0912b84ad971bee5f181a30a Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 28 Oct 2020 11:04:52 -0400 Subject: [PATCH] eja: add two more experimental rank computations. --- mjo/eja/eja_algebra.py | 66 +++++++++++++++++++++++++++++++++++++++++- 1 file changed, 65 insertions(+), 1 deletion(-) diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index fec8a39..34fc0c3 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -794,7 +794,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return tuple( self.random_element() for idx in range(count) ) - def _rank_computation(self): + def _rank_computation1(self): r""" Compute the rank of this algebra using highly suspicious voodoo. @@ -872,6 +872,70 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # answer anyway; when d=3, we go up to x^2. return d-2 + def _rank_computation2(self): + r""" + Instead of using the dimension of an ideal, find the rank of a + matrix containing polynomials. + """ + 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() + + 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(R, n, n, L_x_i_j) + x_powers = [ (vars[k]*(L_x**k)*self.one().to_vector()).row() + for k in range(n) ] + + from sage.matrix.constructor import block_matrix + M = block_matrix(n,1,x_powers) + return M.rank() + + def _rank_computation3(self): + r""" + Similar to :meth:`_rank_computation2`, but it stops echelonizing + as soon as it hits the first zero row. + """ + n = self.dimension() + if n == 0: + return 0 + elif n == 1: + return 1 + + var_names = [ "X" + str(z) for z in range(1,n+1) ] + R = PolynomialRing(self.base_ring(), var_names) + vars = R.gens() + + 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(R, n, n, L_x_i_j) + x_powers = [ vars[k]*(L_x**k)*self.one().to_vector() + for k in range(n) ] + + # Can assume n >= 2 + rows = [x_powers[0]] + M = matrix(rows) + old_rank = 1 + + for d in range(1,n): + rows = M.rows() + [x_powers[d]] + M = matrix(rows) + M.echelonize() + new_rank = M.rank() + if new_rank == old_rank: + return new_rank + else: + old_rank = new_rank + def rank(self): """ Return the rank of this EJA. -- 2.44.2