code: def _to_ZZ_poly(f, ring):
      r"""
      Compute an associate of a polynomial
      `f \in \mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` in
      `\mathbb Z[x_1, \ldots, x_{n-1}][z] / (\check m_{\alpha}(z))[x_0]`,
      where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate
      of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over
      `\mathbb Q`.
  
      Parameters
      ==========
  
      f : PolyElement
          polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]`
      ring : PolyRing
          `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]`
  
      Returns
      =======
  
      f_ : PolyElement
          associate of `f` in
          `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]`
  
      """
      f_ = ring.zero
  
      if isinstance(ring.domain, PolynomialRing):
          domain = ring.domain.domain
      else:
          domain = ring.domain
  
      den = domain.one
  
      for coeff in f.itercoeffs():
          for c in coeff.to_list():
              if c:
                  den = domain.lcm(den, c.denominator)
  
      for monom, coeff in f.iterterms():
          coeff = coeff.to_list()
          m = ring.domain.one
          if isinstance(ring.domain, PolynomialRing):
              m = m.mul_monom(monom[1:])
          n = len(coeff)
  
          for i in range(n):
              if coeff[i]:
                  c = domain.convert(coeff[i] * den) * m
  
                  if (monom[0], n - i - 1) not in f_:
                      f_[(monom[0], n - i - 1)] = c
                  else:
                      f_[(monom[0], n - i - 1)] += c
  
      return f_
  
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 28 for calling a global function with example usage: _to_ZZ_poly(f, ring) and returns: f_