code: def _rational_reconstruction_int_coeffs(hm, m, ring):
      r"""
      Reconstruct every rational coefficient `c_h` of a polynomial `h` in
      `\mathbb Q[t_1, \ldots, t_k][x, z]` from the corresponding integer
      coefficient `c_{h_m}` of a polynomial `h_m` in
      `\mathbb Z[t_1, \ldots, t_k][x, z]` such that
  
      .. math::
  
          c_{h_m} = c_h \; \mathrm{mod} \, m,
  
      where `m \in \mathbb Z`.
  
      The reconstruction is based on the Euclidean Algorithm. In general,
      `m` is not a prime number, so it is possible that this fails for some
      coefficient. In that case ``None`` is returned.
  
      Parameters
      ==========
  
      hm : PolyElement
          polynomial in `\mathbb Z[t_1, \ldots, t_k][x, z]`
      m : Integer
          modulus, not necessarily prime
      ring : PolyRing
          `\mathbb Q[t_1, \ldots, t_k][x, z]`, `h` will be an element of this
          ring
  
      Returns
      =======
  
      h : PolyElement
          reconstructed polynomial in `\mathbb Q[t_1, \ldots, t_k][x, z]` or
          ``None``
  
      See also
      ========
  
      _integer_rational_reconstruction
  
      """
      h = ring.zero
  
      if isinstance(ring.domain, PolynomialRing):
          reconstruction = _rational_reconstruction_int_coeffs
          domain = ring.domain.ring
      else:
          reconstruction = _integer_rational_reconstruction
          domain = hm.ring.domain
  
      for monom, coeff in hm.iterterms():
          coeffh = reconstruction(coeff, m, domain)
  
          if not coeffh:
              return None
  
          h[monom] = coeffh
  
      return h
  
imports: from sympy.core.symbol import Dummy
  from sympy.ntheory import nextprime
  from sympy.ntheory.modular import crt
  from sympy.polys.domains import PolynomialRing
  from sympy.polys.galoistools import gf_gcd, gf_from_dict, gf_gcdex, gf_div, gf_lcm
  from sympy.polys.polyerrors import ModularGCDFailed
  from mpmath import sqrt
  import random
label_desc: Use image node_id 26 for calling a global function with example usage: _rational_reconstruction_int_coeffs(hm, m, ring) and returns: h, None