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We derive the Hasse principle and weak approximation for pencils of certainvarieties in the spirit of work by Colliot-Th\'el\`ene,Sansuc andHarpaz-Skorobogatov-Wittenberg. Our varieties are defined through polynomialsin many variables and part of our work is devoted to establishing Schinzel'shypothesis for polynomials of this kind. This last part is achieved by usingarguments behind Birch's well-known result regarding the Hasse principle forcomplete intersections with the notable difference that we prove our result in50% fewer variables than in the classical Birch setting. We also study theproblem of square-free values of an integer polynomial with 66.6% fewervariables than in the Birch setting.

Rational points and prime values of polynomials in moderately many variables