From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 23 Oct 2020 20:16:58 +0000 (-0400)
Subject: eja: add voodoo _rank_computation() method for EJAs.
X-Git-Url: http://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=47ff028e7a57a9818d9527664efca945acbe6359;p=sage.d.git

eja: add voodoo _rank_computation() method for EJAs.
---

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 0ef8bef..fec8a39 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -794,6 +794,84 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         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
+
     def rank(self):
         """
         Return the rank of this EJA.