We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by Colliot-Thélène–Sansuc and Harpaz–Skorobogatov–Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.

我们根据Colliot-Thélène-Sansuc和Harpaz-Skorobogatov-Wittenberg的工作精神推导了Hasse原理和某些品种纤维化的弱近似。我们的变量是通过多项式在多个变量中定义的，我们的工作的一部分致力于为此类多项式建立Schinzel的假设。这最后一部分是通过使用Birch著名结果中有关完全一致路口的Hasse原理的论点实现的，其显着差异是我们证明了我们的结果比经典Birch设置少了50％的变量。我们还研究了变量比Birch少66.6％的整数多项式的无平方值问题。