code: def fateman_poly_F_3(n):
      """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f)"""
      Y = [Symbol("y_" + str(i)) for i in range(n + 1)]
  
      y_0 = Y[0]
  
      u = Add(*[y ** (n + 1) for y in Y[1:]])
  
      H = Poly((y_0 ** (n + 1) + u + 1) ** 2, *Y)
  
      F = Poly((y_0 ** (n + 1) - u - 2) ** 2, *Y)
      G = Poly((y_0 ** (n + 1) + u + 2) ** 2, *Y)
  
      return H * F, H * G, H
  
imports: from sympy.core import Add, Mul, Symbol, sympify, Dummy, symbols
  from sympy.core.containers import Tuple
  from sympy.core.singleton import S
  from sympy.ntheory import nextprime
  from sympy.polys.densearith import dmp_add_term, dmp_neg, dmp_mul, dmp_sqr
  from sympy.polys.densebasic import dmp_zero, dmp_one, dmp_ground, dup_from_raw_dict, dmp_raise, dup_random
  from sympy.polys.domains import ZZ
  from sympy.polys.factortools import dup_zz_cyclotomic_poly
  from sympy.polys.polyclasses import DMP
  from sympy.polys.polytools import Poly, PurePoly
  from sympy.polys.polyutils import _analyze_gens
  from sympy.utilities import subsets, public, filldedent
  from sympy.polys.rings import ring
label_desc: Use image node_id 10 for calling a global function with example usage: fateman_poly_F_3(n) and returns: unknown,  unknown,  H