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A central question in Arithmetic geometry is to determine for whichpolynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasseprinciple holds for the affine equation $f(t) =N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst extensively studied in theliterature, current results are largely limited to polynomials and numberfields of low degree. In this paper, we establish the Hasse principle for awide family of polynomials and number fields, including polynomials that areproducts of arbitrarily many linear, quadratic or cubic factors. The proofgeneralises an argument of Irving, which makes use of the beta sieve of Rosserand Iwaniec. As a further application of our sieve results, we prove new casesof a conjecture of Harpaz and Wittenberg on locally split values of polynomialsover number fields, and discuss consequences for rational points in fibrations.

Polynomials represented by norm forms via the beta sieve