code: def test_modgcd_bivariate_integers():
      R, x, y = ring("x,y", ZZ)
  
      f, g = R.zero, R.zero
      assert modgcd_bivariate(f, g) == (0, 0, 0)
  
      f, g = 2 * x, R(2)
      assert modgcd_bivariate(f, g) == (2, x, 1)
  
      f, g = x + 2 * y, x + y
      assert modgcd_bivariate(f, g) == (1, f, g)
  
      f, g = x**2 + 2 * x * y + y**2, x**3 + y**3
      assert modgcd_bivariate(f, g) == (
          x + y,
          x + y,
          x**2 - x * y + y**2,
      )
  
      f, g = x * y**2 + 2 * x * y + x, x * y**3 + x
      assert modgcd_bivariate(f, g) == (
          x * y + x,
          y + 1,
          y**2 - y + 1,
      )
  
      f, g = x**2 * y**2 + x**2 * y + 1, x * y**2 + x * y + 1
      assert modgcd_bivariate(f, g) == (1, f, g)
  
      f = 2 * x * y**2 + 4 * x * y + 2 * x + y**2 + 2 * y + 1
      g = 2 * x * y**3 + 2 * x + y**3 + 1
      assert modgcd_bivariate(f, g) == (
          2 * x * y + 2 * x + y + 1,
          y + 1,
          y**2 - y + 1,
      )
  
      f, g = 2 * x**2 + 4 * x + 2, x + 1
      assert modgcd_bivariate(f, g) == (x + 1, 2 * x + 2, 1)
  
      f, g = x + 1, 2 * x**2 + 4 * x + 2
      assert modgcd_bivariate(f, g) == (x + 1, 1, 2 * x + 2)
  
      f = 2 * x**2 + 4 * x * y - 2 * x - 4 * y
      g = x**2 + x - 2
      assert modgcd_bivariate(f, g) == (x - 1, 2 * x + 4 * y, x + 2)
  
      f = 2 * x**2 + 2 * x * y - 3 * x - 3 * y
      g = 4 * x * y - 2 * x + 4 * y**2 - 2 * y
      assert modgcd_bivariate(f, g) == (x + y, 2 * x - 3, 4 * y - 2)
  
file_path: sympy/sympy/polys/tests/test_modulargcd.py
imports: from sympy.polys.rings import ring
  from sympy.polys.domains import ZZ, QQ, AlgebraicField
  from sympy.polys.modulargcd import modgcd_univariate, modgcd_bivariate, _chinese_remainder_reconstruction_multivariate, modgcd_multivariate, _to_ZZ_poly, _to_ANP_poly, func_field_modgcd, _func_field_modgcd_m
  from sympy.functions.elementary.miscellaneous import sqrt
label_desc: Use image node_id 2 for calling a global function with example usage: test_modgcd_bivariate_integers() without return types