  We derive the Hasse principle and weak approximation for pencils of certain
varieties in the spirit of work by Colliot-Th\'el\`ene,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.
