X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=ff2b5d7a9c52ff5c13df7d7aa3682899b9a30559;hb=cd164e717e7224ec1af16d31152555f0c7cf49cf;hp=fec8a39f00840c10ee22a493756e75abcc6da317;hpb=47ff028e7a57a9818d9527664efca945acbe6359;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index fec8a39..ff2b5d7 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -791,97 +791,21 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in range(count) ) - - - def _rank_computation(self): - r""" - Compute the rank of this algebra using highly suspicious voodoo. - - ALGORITHM: - - We first compute the basis representation of the operator L_x - using polynomial indeterminates are placeholders for the - coordinates of "x", which is arbitrary. We then use that - matrix to compute the (polynomial) entries of x^0, x^1, ..., - x^d,... for increasing values of "d", starting at zero. The - idea is that. If we also add "coefficient variables" a_0, - a_1,... to the ring, we can form the linear combination - a_0*x^0 + ... + a_d*x^d = 0, and ask what dimension the - solution space has as an affine variety. When "d" is smaller - than the rank, we expect that dimension to be the number of - coordinates of "x", since we can set *those* to whatever we - want, but linear independence forces the coefficients a_i to - be zero. Eventually, when "d" passes the rank, the dimension - of the solution space begins to grow, because we can *still* - set the coordinates of "x" arbitrarily, but now there are some - coefficients that make the sum zero as well. So, when the - dimension of the variety jumps, we return the corresponding - "d" as the rank of the algebra. This appears to work. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (HadamardEJA, - ....: JordanSpinEJA, - ....: RealSymmetricEJA, - ....: ComplexHermitianEJA, - ....: QuaternionHermitianEJA) - - EXAMPLES:: - - sage: J = HadamardEJA(5) - sage: J._rank_computation() == J.rank() - True - sage: J = JordanSpinEJA(5) - sage: J._rank_computation() == J.rank() - True - sage: J = RealSymmetricEJA(4) - sage: J._rank_computation() == J.rank() - True - sage: J = ComplexHermitianEJA(3) - sage: J._rank_computation() == J.rank() - True - sage: J = QuaternionHermitianEJA(2) - sage: J._rank_computation() == J.rank() - True - - """ - n = self.dimension() - var_names = [ "X" + str(z) for z in range(1,n+1) ] - d = 0 - ideal_dim = len(var_names) - 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[d+k]*self.monomial(k).operator().matrix()[i,j] - for k in range(n) ) - - while ideal_dim == len(var_names): - coeff_names = [ "a" + str(z) for z in range(d) ] - R = PolynomialRing(self.base_ring(), coeff_names + var_names) - vars = R.gens() - 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(d) ] - eqs = [ sum(x_powers[k][j] for k in range(d)) for j in range(n) ] - ideal_dim = R.ideal(eqs).dimension() - d += 1 - - # Subtract one because we increment one too many times, and - # subtract another one because "d" is one greater than the - # answer anyway; when d=3, we go up to x^2. - return d-2 + return tuple( self.random_element() for idx in range(count) ) + @cached_method def rank(self): """ Return the rank of this EJA. ALGORITHM: - The author knows of no algorithm to compute the rank of an EJA - where only the multiplication table is known. In lieu of one, we - require the rank to be specified when the algebra is created, - and simply pass along that number here. + We first compute the polynomial "column matrices" `p_{k}` that + evaluate to `x^k` on the coordinates of `x`. Then, we begin + adding them to a matrix one at a time, and trying to solve the + system that makes `p_{0}`,`p_{1}`,..., `p_{s-1}` add up to + `p_{s}`. This will succeed only when `s` is the rank of the + algebra, as proven in a recent draft paper of mine. SETUP:: @@ -926,8 +850,80 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: r > 0 or (r == 0 and J.is_trivial()) True + Ensure that computing the rank actually works, since the ranks + of all simple algebras are known and will be cached by default:: + + sage: J = HadamardEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 4 + + :: + + sage: J = JordanSpinEJA(4) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + + :: + + sage: J = RealSymmetricEJA(3) + sage: J.rank.clear_cache() + sage: J.rank() + 3 + + :: + + sage: J = ComplexHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + + :: + + sage: J = QuaternionHermitianEJA(2) + sage: J.rank.clear_cache() + sage: J.rank() + 2 + """ - return self._rank + 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 + M = matrix([x_powers[0]]) + old_rank = 1 + + for d in range(1,n): + M = matrix(M.rows() + [x_powers[d]]) + M.echelonize() + # TODO: we've basically solved the system here. + # We should save the echelonized matrix somehow + # so that it can be reused in the charpoly method. + new_rank = M.rank() + if new_rank == old_rank: + return new_rank + else: + old_rank = new_rank + + return n def vector_space(self): @@ -1147,6 +1143,28 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): **kwargs) + def _rank_computation(self): + r""" + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + if self._basis_normalizers is None: + # We didn't normalize, so assume that the basis we started + # with had entries in a nice field. + return super(MatrixEuclideanJordanAlgebra, self)._rank_computation() + else: + basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + self._basis_normalizers) ) + + # Do this over the rationals and convert back at the end. + # Only works because we know the entries of the basis are + # integers. + J = MatrixEuclideanJordanAlgebra(QQ, + basis, + self.rank(), + normalize_basis=False) + return J._rank_computation() + @cached_method def _charpoly_coeff(self, i): """