Rational solutions of certain Diophantine equations involving norms

• Autorzy: Maciej Ulas
• Języki publikacji: EN

Abstrakty

EN

We present some results concerning the unirationality of the algebraic variety $𝓢_{f}$ given by the equation
$N_{K/k} (X₁ + αX₂ + α²X₃) = f(t)$,
where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and $𝓢_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $𝓢_{f}$ is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $𝓢_{f}$ (with a non-trivial k-rational point) is proved for any polynomial f of degree 6 with f not equivalent to a polynomial h satisfying h(t) = h(ζ₃t), where ζ₃ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by a root of the polynomial h(x) = x³ +ax + b ∈ k[x], provided that f(t) = t⁶ + a₄t⁴ + a₁t + a₀ ∈ k[t] with a₁a₄ ≠ 0.

Kategorie tematyczne

• 11D85: Representation problems
• 11D57: Multiplicative and norm form equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 165
• Numer: 1
• Strony: 47-56

Daty

wydano

2014

Twórcy

autor

Maciej Ulas

• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Identyfikatory

DOI

10.4064/aa165-1-3