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POLYNOMIALS REPRESENTED BY NORM FORMS VIA THE BETA SIEVE
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 24 / Issue 4 / July 2025
- Published online by Cambridge University Press:
- 25 February 2025, pp. 1463-1519
- Print publication:
- July 2025
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- Article
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A central question in Arithmetic geometry is to determine for which polynomials
$f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation 
$f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.
