+ def _a_regular_element(self):
+ """
+ Guess a regular element. Needed to compute the basis for our
+ characteristic polynomial coefficients.
+ """
+ gs = self.gens()
+ z = self.sum( (i+1)*gs[i] for i in range(len(gs)) )
+ if not z.is_regular():
+ raise ValueError("don't know a regular element")
+ return z
+
+
+ @cached_method
+ def _charpoly_basis_space(self):
+ """
+ Return the vector space spanned by the basis used in our
+ characteristic polynomial coefficients. This is used not only to
+ compute those coefficients, but also any time we need to
+ evaluate the coefficients (like when we compute the trace or
+ determinant).
+ """
+ z = self._a_regular_element()
+ V = z.vector().parent().ambient_vector_space()
+ V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) )
+ b = (V1.basis() + V1.complement().basis())
+ return V.span_of_basis(b)
+
+
+ @cached_method
+ def _charpoly_coeff(self, i):
+ """
+ Return the coefficient polynomial "a_{i}" of this algebra's
+ general characteristic polynomial.
+
+ Having this be a separate cached method lets us compute and
+ store the trace/determinant (a_{r-1} and a_{0} respectively)
+ separate from the entire characteristic polynomial.
+ """
+ (A_of_x, x, xr, detA) = self._charpoly_matrix_system()
+ R = A_of_x.base_ring()
+ if i >= self.rank():
+ # Guaranteed by theory
+ return R.zero()
+
+ # Danger: the in-place modification is done for performance
+ # reasons (reconstructing a matrix with huge polynomial
+ # entries is slow), but I don't know how cached_method works,
+ # so it's highly possible that we're modifying some global
+ # list variable by reference, here. In other words, you
+ # probably shouldn't call this method twice on the same
+ # algebra, at the same time, in two threads
+ Ai_orig = A_of_x.column(i)
+ A_of_x.set_column(i,xr)
+ numerator = A_of_x.det()
+ A_of_x.set_column(i,Ai_orig)
+
+ # We're relying on the theory here to ensure that each a_i is
+ # indeed back in R, and the added negative signs are to make
+ # the whole charpoly expression sum to zero.
+ return R(-numerator/detA)
+
+
+ @cached_method
+ def _charpoly_matrix_system(self):
+ """
+ Compute the matrix whose entries A_ij are polynomials in
+ X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
+ corresponding to `x^r` and the determinent of the matrix A =
+ [A_ij]. In other words, all of the fixed (cachable) data needed
+ to compute the coefficients of the characteristic polynomial.
+ """
+ r = self.rank()
+ n = self.dimension()
+
+ # Construct a new algebra over a multivariate polynomial ring...
+ names = ['X' + str(i) for i in range(1,n+1)]
+ R = PolynomialRing(self.base_ring(), names)
+ J = FiniteDimensionalEuclideanJordanAlgebra(R,
+ self._multiplication_table,
+ rank=r)
+
+ idmat = identity_matrix(J.base_ring(), n)
+
+ W = self._charpoly_basis_space()
+ W = W.change_ring(R.fraction_field())
+
+ # Starting with the standard coordinates x = (X1,X2,...,Xn)
+ # and then converting the entries to W-coordinates allows us
+ # to pass in the standard coordinates to the charpoly and get
+ # back the right answer. Specifically, with x = (X1,X2,...,Xn),
+ # we have
+ #
+ # W.coordinates(x^2) eval'd at (standard z-coords)
+ # =
+ # W-coords of (z^2)
+ # =
+ # W-coords of (standard coords of x^2 eval'd at std-coords of z)
+ #
+ # We want the middle equivalent thing in our matrix, but use
+ # the first equivalent thing instead so that we can pass in
+ # standard coordinates.
+ x = J(vector(R, R.gens()))
+ l1 = [column_matrix(W.coordinates((x**k).vector())) for k in range(r)]
+ l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)]
+ A_of_x = block_matrix(R, 1, n, (l1 + l2))
+ xr = W.coordinates((x**r).vector())
+ return (A_of_x, x, xr, A_of_x.det())
+
+
+ @cached_method
+ def characteristic_polynomial(self):
+ """
+
+ .. WARNING::
+
+ This implementation doesn't guarantee that the polynomial
+ denominator in the coefficients is not identically zero, so
+ theoretically it could crash. The way that this is handled
+ in e.g. Faraut and Koranyi is to use a basis that guarantees
+ the denominator is non-zero. But, doing so requires knowledge
+ of at least one regular element, and we don't even know how
+ to do that. The trade-off is that, if we use the standard basis,
+ the resulting polynomial will accept the "usual" coordinates. In
+ other words, we don't have to do a change of basis before e.g.
+ computing the trace or determinant.
+
+ EXAMPLES:
+
+ The characteristic polynomial in the spin algebra is given in
+ Alizadeh, Example 11.11::
+
+ sage: J = JordanSpinEJA(3)
+ sage: p = J.characteristic_polynomial(); p
+ X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
+ sage: xvec = J.one().vector()
+ sage: p(*xvec)
+ t^2 - 2*t + 1
+
+ """
+ r = self.rank()
+ n = self.dimension()
+
+ # The list of coefficient polynomials a_1, a_2, ..., a_n.
+ a = [ self._charpoly_coeff(i) for i in range(n) ]
+
+ # We go to a bit of trouble here to reorder the
+ # indeterminates, so that it's easier to evaluate the
+ # characteristic polynomial at x's coordinates and get back
+ # something in terms of t, which is what we want.
+ R = a[0].parent()
+ S = PolynomialRing(self.base_ring(),'t')
+ t = S.gen(0)
+ S = PolynomialRing(S, R.variable_names())
+ t = S(t)
+
+ # Note: all entries past the rth should be zero. The
+ # coefficient of the highest power (x^r) is 1, but it doesn't
+ # appear in the solution vector which contains coefficients
+ # for the other powers (to make them sum to x^r).
+ if (r < n):
+ a[r] = 1 # corresponds to x^r
+ else:
+ # When the rank is equal to the dimension, trying to
+ # assign a[r] goes out-of-bounds.
+ a.append(1) # corresponds to x^r
+
+ return sum( a[k]*(t**k) for k in range(len(a)) )
+
+
+ def inner_product(self, x, y):
+ """
+ The inner product associated with this Euclidean Jordan algebra.
+
+ Defaults to the trace inner product, but can be overridden by
+ subclasses if they are sure that the necessary properties are
+ satisfied.
+
+ EXAMPLES:
+
+ The inner product must satisfy its axiom for this algebra to truly
+ be a Euclidean Jordan Algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
+ """
+ if (not x in self) or (not y in self):
+ raise TypeError("arguments must live in this algebra")
+ return x.trace_inner_product(y)
+
+