text: """
  Local Normal Form
  """
  
  # ****************************************************************************
  #       Copyright (C) 2007 William Stein and Jonathan Hanke
  #
  #  Distributed under the terms of the GNU General Public License (GPL)
  #
  #    This code is distributed in the hope that it will be useful,
  #    but WITHOUT ANY WARRANTY; without even the implied warranty of
  #    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  #    General Public License for more details.
  #
  #  The full text of the GPL is available at:
  #
  #                  https://www.gnu.org/licenses/
  # ****************************************************************************
  
  import copy
  from sage.rings.infinity import Infinity
  from sage.rings.integer_ring import IntegerRing, ZZ
  from sage.rings.rational_field import QQ
  from sage.arith.all import GCD, valuation, is_prime
  
  
  def find_entry_with_minimal_scale_at_prime(self, p):
      """
      Finds the entry of the quadratic form with minimal scale at the
      prime p, preferring diagonal entries in case of a tie.  (I.e.  If
      we write the quadratic form as a symmetric matrix M, then this
      entry M[i,j] has the minimal valuation at the prime p.)
  
      Note: This answer is independent of the kind of matrix (Gram or
      Hessian) associated to the form.
  
      INPUT:
  
          `p` -- a prime number > 0
  
      OUTPUT:
  
          a pair of integers >= 0
  
      EXAMPLES::
  
          sage: Q = QuadraticForm(ZZ, 2, [6, 2, 20]); Q
          Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 6 2 ]
          [ * 20 ]
          sage: Q.find_entry_with_minimal_scale_at_prime(2)
          (0, 1)
          sage: Q.find_entry_with_minimal_scale_at_prime(3)
          (1, 1)
          sage: Q.find_entry_with_minimal_scale_at_prime(5)
          (0, 0)
  
      """
      n = self.dim()
      min_val = Infinity
      ij_index = None
      val_2 = valuation(2, p)
      for d in range(n):           # d = difference j-i
          for e in range(n - d):    # e is the length of the diagonal with value d.
  
              # Compute the valuation of the entry
              if d == 0:
                  tmp_val = valuation(self[e, e + d], p)
              else:
                  tmp_val = valuation(self[e, e + d], p) - val_2
  
              # Check if it's any smaller than what we have
              if tmp_val < min_val:
                  ij_index = (e, e + d)
                  min_val = tmp_val
  
      # Return the result
      return ij_index
  
  
  def local_normal_form(self, p):
      r"""
      Return a locally integrally equivalent quadratic form over
      the `p`-adic integers `\ZZ_p` which gives the Jordan decomposition.
  
      The Jordan components are written as sums of blocks of size <= 2
      and are arranged by increasing scale, and then by increasing norm.
      This is equivalent to saying that we put the 1x1 blocks before
      the 2x2 blocks in each Jordan component.
  
      INPUT:
  
      - `p` -- a positive prime number.
  
      OUTPUT:
  
      a quadratic form over `\ZZ`
  
      .. WARNING::
  
          Currently this only works for quadratic forms defined over `\ZZ`.
  
      EXAMPLES::
  
          sage: Q = QuadraticForm(ZZ, 2, [10,4,1])
          sage: Q.local_normal_form(5)
          Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 1 0 ]
          [ * 6 ]
  
      ::
  
          sage: Q.local_normal_form(3)
          Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 10 0 ]
          [ * 15 ]
  
          sage: Q.local_normal_form(2)
          Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 1 0 ]
          [ * 6 ]
      """
      # Sanity Checks
      if (self.base_ring() != IntegerRing()):
          raise NotImplementedError("Oops!  This currently only works for quadratic forms defined over IntegerRing(). =(")
      if not ((p>=2) and is_prime(p)):
          raise TypeError("Oops!  p is not a positive prime number. =(")
  
      # Some useful local variables
      Q = self.parent()(self.base_ring(), self.dim(), self.coefficients())
  
      # Prepare the final form to return
      Q_Jordan = copy.deepcopy(self)
      Q_Jordan.__init__(self.base_ring(), 0)
  
      while Q.dim() > 0:
          n = Q.dim()
  
          # Step 1: Find the minimally p-divisible matrix entry, preferring diagonals
          # -------------------------------------------------------------------------
          (min_i, min_j) = Q.find_entry_with_minimal_scale_at_prime(p)
          if min_i == min_j:
              min_val = valuation(2 * Q[min_i, min_j], p)
          else:
              min_val = valuation(Q[min_i, min_j], p)
  
          # Error if we still haven't seen non-zero coefficients!
          if (min_val == Infinity):
              raise RuntimeError("Oops!  The original matrix is degenerate. =(")
  
          # Step 2: Arrange for the upper leftmost entry to have minimal valuation
          # ----------------------------------------------------------------------
          if (min_i == min_j):
              block_size = 1
              Q.swap_variables(0, min_i, in_place = True)
          else:
              # Work in the upper-left 2x2 block, and replace it by its 2-adic equivalent form
              Q.swap_variables(0, min_i, in_place = True)
              Q.swap_variables(1, min_j, in_place = True)
  
              # 1x1 => make upper left the smallest
              if (p != 2):
                  block_size = 1
                  Q.add_symmetric(1, 0, 1, in_place=True)
              # 2x2 => replace it with the appropriate 2x2 matrix
              else:
                  block_size = 2
  
          # Step 3: Clear out the remaining entries
          #  ---------------------------------------
          min_scale = p ** min_val                             # This is the minimal valuation of the Hessian matrix entries.
  
          # Perform cancellation over Z by ensuring divisibility
          if (block_size == 1):
              a = 2 * Q[0,0]
              for j in range(block_size, n):
                  b = Q[0, j]
                  g = GCD(a, b)
  
                  # Sanity Check:  a/g is a p-unit
                  if valuation(g, p) != valuation(a, p):
                      raise RuntimeError("Oops!  We have a problem with our rescaling not preserving p-integrality!")
  
                  Q.multiply_variable(ZZ(a/g), j, in_place = True)   # Ensures that the new b entry is divisible by a
                  Q.add_symmetric(ZZ(-b/g), j, 0, in_place = True)  # Performs the cancellation
  
          elif (block_size == 2):
              a1 = 2 * Q[0,0]
              a2 = Q[0, 1]
              b1 = Q[1, 0]      # This is the same as a2
              b2 = 2 * Q[1, 1]
  
              big_det = (a1*b2 - a2*b1)
              small_det = big_det / (min_scale * min_scale)
  
              # Cancels out the rows/columns of the 2x2 block
              for j in range(block_size, n):
                  a = Q[0, j]
                  b = Q[1, j]
  
                  # Ensures an integral result (scale jth row/column by big_det)
                  Q.multiply_variable(big_det, j, in_place = True)
  
                  # Performs the cancellation (by producing -big_det * jth row/column)
                  Q.add_symmetric(ZZ(-(a*b2 - b*a2)), j, 0, in_place = True)
                  Q.add_symmetric(ZZ(-(-a*b1 + b*a1)), j, 1, in_place = True)
  
                  # Now remove the extra factor (non p-unit factor) in big_det we introduced above
                  Q.divide_variable(ZZ(min_scale * min_scale), j, in_place = True)
  
              # Uses Cassels's proof to replace the remaining 2 x 2 block
              if (((1 + small_det) % 8) == 0):
                  Q[0, 0] = 0
                  Q[1, 1] = 0
                  Q[0, 1] = min_scale
              elif (((5 + small_det) % 8) == 0):
                  Q[0, 0] = min_scale
                  Q[1, 1] = min_scale
                  Q[0, 1] = min_scale
              else:
                  raise RuntimeError("Error in LocalNormal: Impossible behavior for a 2x2 block! \n")
  
          # Check that the cancellation worked, extract the upper-left block, and trim Q to handle the next block.
          for i in range(block_size):
              for j in range(block_size, n):
                  if Q[i,j] != 0:
                      raise RuntimeError("Oops!  The cancellation didn't work properly at entry (" + str(i) + ", " + str(j) + ").")
          Q_Jordan = Q_Jordan + Q.extract_variables(range(block_size))
          Q = Q.extract_variables(range(block_size, n))
  
      return Q_Jordan
  
  
  def jordan_blocks_by_scale_and_unimodular(self, p, safe_flag=True):
      r"""
      Return a list of pairs `(s_i, L_i)` where `L_i` is a maximal
      `p^{s_i}`-unimodular Jordan component which is further decomposed into
      block diagonals of block size `\le 2`.
  
      For each `L_i` the 2x2 blocks are listed after the 1x1 blocks
      (which follows from the convention of the
      :meth:`local_normal_form` method).
  
      .. NOTE::
  
          The decomposition of each `L_i` into smaller blocks is not unique!
  
      The ``safe_flag`` argument allows us to select whether we want a copy of
      the output, or the original output.  By default ``safe_flag = True``, so we
      return a copy of the cached information.  If this is set to ``False``, then
      the routine is much faster but the return values are vulnerable to being
      corrupted by the user.
  
      INPUT:
  
      - `p` -- a prime number > 0.
  
      OUTPUT:
  
      A list of pairs `(s_i, L_i)` where:
  
      - `s_i` is an integer,
      - `L_i` is a block-diagonal unimodular quadratic form over `\ZZ_p`.
  
      .. note::
  
          These forms `L_i` are defined over the `p`-adic integers, but by a
          matrix over `\ZZ` (or `\QQ`?).
  
      EXAMPLES::
  
          sage: Q = DiagonalQuadraticForm(ZZ, [1,9,5,7])
          sage: Q.jordan_blocks_by_scale_and_unimodular(3)
          [(0, Quadratic form in 3 variables over Integer Ring with coefficients:
          [ 1 0 0 ]
          [ * 5 0 ]
          [ * * 7 ]), (2, Quadratic form in 1 variables over Integer Ring with coefficients:
          [ 1 ])]
  
      ::
  
          sage: Q2 = QuadraticForm(ZZ, 2, [1,1,1])
          sage: Q2.jordan_blocks_by_scale_and_unimodular(2)
          [(-1, Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 2 2 ]
          [ * 2 ])]
          sage: Q = Q2 + Q2.scale_by_factor(2)
          sage: Q.jordan_blocks_by_scale_and_unimodular(2)
          [(-1, Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 2 2 ]
          [ * 2 ]), (0, Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 2 2 ]
          [ * 2 ])]
      """
      # Try to use the cached result
      try:
          if safe_flag:
              return copy.deepcopy(self.__jordan_blocks_by_scale_and_unimodular_dict[p])
          else:
              return self.__jordan_blocks_by_scale_and_unimodular_dict[p]
      except Exception:
          # Initialize the global dictionary if it doesn't exist
          if not hasattr(self, '__jordan_blocks_by_scale_and_unimodular_dict'):
              self.__jordan_blocks_by_scale_and_unimodular_dict = {}
  
  
      # Deal with zero dim'l forms
      if self.dim() == 0:
          return []
  
  
      # Find the Local Normal form of Q at p
      Q1 = self.local_normal_form(p)
  
  
      # Parse this into Jordan Blocks
      n = Q1.dim()
      tmp_Jordan_list = []
      i = 0
      start_ind = 0
      if (n >= 2) and (Q1[0,1] != 0):
          start_scale = valuation(Q1[0,1], p) - 1
      else:
          start_scale = valuation(Q1[0,0], p)
  
      while (i < n):
  
          # Determine the size of the current block
          if (i == n-1) or (Q1[i,i+1] == 0):
              block_size = 1
          else:
              block_size = 2
  
          # Determine the valuation of the current block
          if block_size == 1:
              block_scale = valuation(Q1[i,i], p)
          else:
              block_scale = valuation(Q1[i,i+1], p) - 1
  
          # Process the previous block if the valuation increased
          if block_scale > start_scale:
              tmp_Jordan_list += [(start_scale, Q1.extract_variables(range(start_ind, i)).scale_by_factor(ZZ(1) / (QQ(p)**(start_scale))))]
              start_ind = i
              start_scale = block_scale
  
          # Increment the index
          i += block_size
  
      # Add the last block
      tmp_Jordan_list += [(start_scale, Q1.extract_variables(range(start_ind, n)).scale_by_factor(ZZ(1) / QQ(p)**(start_scale)))]
  
  
      # Cache the result
      self.__jordan_blocks_by_scale_and_unimodular_dict[p] = tmp_Jordan_list
  
      # Return the result
      return tmp_Jordan_list
  
  
  def jordan_blocks_in_unimodular_list_by_scale_power(self, p):
      """
      Returns a list of Jordan components, whose component at index i
      should be scaled by the factor p^i.
  
      This is only defined for integer-valued quadratic forms
      (i.e. forms with base_ring ZZ), and the indexing only works
      correctly for p=2 when the form has an integer Gram matrix.
  
      INPUT:
  
          self -- a quadratic form over ZZ, which has integer Gram matrix if p == 2
          `p` -- a prime number > 0
  
      OUTPUT:
  
          a list of p-unimodular quadratic forms
  
      EXAMPLES::
  
          sage: Q = QuadraticForm(ZZ, 3, [2, -2, 0, 3, -5, 4])
          sage: Q.jordan_blocks_in_unimodular_list_by_scale_power(2)
          Traceback (most recent call last):
          ...
          TypeError: Oops!  The given quadratic form has a Jordan component with a negative scale exponent!
          This routine requires an integer-matrix quadratic form for the output indexing to work properly!
  
          sage: Q.scale_by_factor(2).jordan_blocks_in_unimodular_list_by_scale_power(2)
          [Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 0 2 ]
          [ * 0 ], Quadratic form in 0 variables over Integer Ring with coefficients:
          , Quadratic form in 1 variables over Integer Ring with coefficients:
          [ 345 ]]
  
          sage: Q.jordan_blocks_in_unimodular_list_by_scale_power(3)
          [Quadratic form in 2 variables over Integer Ring with coefficients:
          [ 2 0 ]
          [ * 10 ], Quadratic form in 1 variables over Integer Ring with coefficients:
          [ 2 ]]
      """
      # Sanity Check
      if self.base_ring() != ZZ:
          raise TypeError("Oops!  This method only makes sense for integer-valued quadratic forms (i.e. defined over ZZ).")
  
      # Deal with zero dim'l forms
      if self.dim() == 0:
          return []
  
      # Find the Jordan Decomposition
      list_of_jordan_pairs = self.jordan_blocks_by_scale_and_unimodular(p)
      scale_list = [P[0] for P in list_of_jordan_pairs]
      s_max = max(scale_list)
      if min(scale_list) < 0:
          raise TypeError("Oops!  The given quadratic form has a Jordan component with a negative scale exponent!\n"
          + "This routine requires an integer-matrix quadratic form for the output indexing to work properly!")
  
      # Make the new list of unimodular Jordan components
      zero_form = self.parent()(ZZ, 0)
      list_by_scale = [zero_form for _ in range(s_max + 1)]
      for P in list_of_jordan_pairs:
          list_by_scale[P[0]] = P[1]
  
      # Return the new list
      return list_by_scale
  