Cubic polynomials represented by norm forms

Abstract

Abstract In the present paper, we show that for an irreducible cubic f ∈ ℤ ⁢ [ x ] ${f\in\mathbb{Z}[x]}$ and a full norm form 𝐍 ⁢ ( x 1 , … , x k ) ${\mathbf{N}(x_{1},\ldots,x_{k})}$ for a number field K / ℚ ${K/\mathbb{Q}}$ satisfying certain hypotheses the variety f ⁢ ( t ) = 𝐍 ⁢ ( x 1 , … , x k ) ≠ 0 $f(t)=\mathbf{N}(x_{1},\ldots,x_{k})\neq 0$ satisfies the Hasse principle. Our proof uses sieve methods.