text: from .PyPolyBoRi import (Monomial, random_set, Polynomial,
                           set_random_seed, ll_red_nf_redsb, Variable)
  from .ll import ll_encode
  from random import Random
  from pprint import pformat
  from .blocks import declare_ring
  
  
  def gen_random_poly(ring, l, deg, vars_set, seed=123):
      """
      Generate a random polynomial with coefficients in ``ring``.
  
      EXAMPLES::
  
          sage: from sage.rings.polynomial.pbori.PyPolyBoRi import Ring, Variable
          sage: from sage.rings.polynomial.pbori.randompoly import gen_random_poly
          sage: r = Ring(16)
          sage: vars = [Variable(i,r) for i in range(10)]
          sage: gen_random_poly(r, 4, 10, vars)  # random
          x(0)*x(1)*x(2)*x(5)*x(8)*x(9) + x(0)*x(1)*x(4)*x(6) + x(0)*x(2)*x(3)*x(7)*x(9) + x(5)*x(8)
      """
      myrange = vars_set
      r = Random(seed)
  
      def helper(samples):
          if samples == 0:
              return Polynomial(ring.zero())
          if samples == 1:
              d = r.randint(0, deg)
              variables = r.sample(myrange, d)
              m = Monomial(ring)
              for v in sorted(set(variables), reverse=True):
                  m = m * Variable(v, ring)
              return Polynomial(m)
          assert samples >= 2
          return helper(samples // 2) + helper(samples - samples // 2)
      p = Polynomial(ring.zero())
      while len(p) < l:
          p = Polynomial(p.set().union(helper(l - len(p)).set()))
      return p
  
  
  def sparse_random_system(ring, number_of_polynomials, variables_per_polynomial,
                           degree, random_seed=None):
      r"""
      Generate a sparse random system.
  
      Generate a system, which is sparse in the sense, that each polynomial
      contains only a small subset of variables. In each variable that occurrs
      in a polynomial it is dense in the terms up to the given degree
      (every term occurs with probability 1/2).
  
      The system will be satisfiable by at least one solution.
  
      TESTS::
  
          sage: from sage.rings.polynomial.pbori import Ring, groebner_basis
          sage: r = Ring(10)
          sage: from sage.rings.polynomial.pbori.randompoly import sparse_random_system
          sage: s = sparse_random_system(r, number_of_polynomials=20, variables_per_polynomial=3, degree=2, random_seed=int(123))
          sage: [p.deg() for p in s]
          [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
          sage: sorted(groebner_basis(s), reverse=True)
          [x(0), x(1) + 1, x(2), x(3) + 1, x(4) + 1, x(5), x(6), x(7) + 1, x(8) + 1, x(9) + 1]
      """
      if random_seed is not None:
          set_random_seed(random_seed)
      random_generator = Random(random_seed)
      solutions = []
      variables = [ring.variable(i) for i in range(ring.n_variables())]
      for v in variables:
          solutions.append(v + random_generator.randint(0, 1))
      solutions = ll_encode(solutions)
      res = []
      while len(res) < number_of_polynomials:
          variables_as_monomial = Monomial(
              random_generator.sample(
                  variables,
                  variables_per_polynomial)
          )
          p = Polynomial(random_set(variables_as_monomial, 2 ** (
              variables_per_polynomial - 1)))
          p = sum([p.graded_part(i) for i in range(degree + 1)])
          if p.deg() == degree:
              res.append(p)
      # evaluate it to guarantee a solution
      return [p + ll_red_nf_redsb(p, solutions) for p in res]
  
  
  
  
  def sparse_random_system_data_file_content(number_of_variables, **kwds):
      r"""
      TESTS::
  
          sage: from sage.rings.polynomial.pbori.randompoly import sparse_random_system_data_file_content
          sage: sparse_random_system_data_file_content(10, number_of_polynomials=5, variables_per_polynomial=3, degree=2, random_seed=int(123))
          "declare_ring(['x'+str(i) for in range(10)])\nideal=\\\n[...]\n\n"
      """
      dummy_dict = {}
      r = declare_ring(['x' + str(i) for i in range(number_of_variables)],
                       dummy_dict)
      polynomials = sparse_random_system(r, **kwds)
      polynomials = pformat(polynomials)
      return "declare_ring(['x'+str(i) for in range(%s)])\nideal=\\\n%s\n\n" % (
          number_of_variables, polynomials)
  