code: def _rational_reconstruction_func_coeffs(hm, p, m, ring, k):
      r"""
      Reconstruct every coefficient `c_h` of a polynomial `h` in
      `\mathbb Z_p(t_k)[t_1, \ldots, t_{k-1}][x, z]` from the corresponding
      coefficient `c_{h_m}` of a polynomial `h_m` in
      `\mathbb Z_p[t_1, \ldots, t_k][x, z] \cong \mathbb Z_p[t_k][t_1, \ldots, t_{k-1}][x, z]`
      such that
  
      .. math::
  
          c_{h_m} = c_h \; \mathrm{mod} \, m,
  
      where `m \in \mathbb Z_p[t]`.
  
      The reconstruction is based on the Euclidean Algorithm. In general, `m`
      is not irreducible, 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]`
      p : Integer
          prime number, modulus of `\mathbb Z_p`
      m : PolyElement
          modulus, polynomial in `\mathbb Z[t]`, not necessarily irreducible
      ring : PolyRing
          `\mathbb Z(t_k)[t_1, \ldots, t_{k-1}][x, z]`, `h` will be an
          element of this ring
      k : Integer
          index of the parameter `t_k` which will be reconstructed
  
      Returns
      =======
  
      h : PolyElement
          reconstructed polynomial in
          `\mathbb Z(t_k)[t_1, \ldots, t_{k-1}][x, z]` or ``None``
  
      See also
      ========
  
      _rational_function_reconstruction
  
      """
      h = ring.zero
  
      for monom, coeff in hm.iterterms():
          if k == 0:
              coeffh = _rational_function_reconstruction(coeff, p, m)
  
              if not coeffh:
                  return None
  
          else:
              coeffh = ring.domain.zero
              for mon, c in coeff.drop_to_ground(k).iterterms():
                  ch = _rational_function_reconstruction(c, p, m)
  
                  if not ch:
                      return None
  
                  coeffh[mon] = ch
  
          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 18 for calling a global function with example usage: _rational_reconstruction_func_coeffs(hm, p, m, ring, k) and returns: h, None, None