Rational solutions of certain Diophantine equations involving norms

Maciej Ulas. "Rational solutions of certain Diophantine equations involving norms." Acta Arithmetica 165.1 (2014): 47-56. <http://eudml.org/doc/279029>.

@article{MaciejUlas2014,
abstract = {We present some results concerning the unirationality of the algebraic variety $_\{f\}$ 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 $_\{f\}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_\{f\}$ 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 $_\{f\}$ (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.},
author = {Maciej Ulas},
journal = {Acta Arithmetica},
keywords = {rational points; châtelet threefold; unirationality; norm form; cubic extension},
language = {eng},
number = {1},
pages = {47-56},
title = {Rational solutions of certain Diophantine equations involving norms},
url = {http://eudml.org/doc/279029},
volume = {165},
year = {2014},
}

TY - JOUR
AU - Maciej Ulas
TI - Rational solutions of certain Diophantine equations involving norms
JO - Acta Arithmetica
PY - 2014
VL - 165
IS - 1
SP - 47
EP - 56
AB - We present some results concerning the unirationality of the algebraic variety $_{f}$ 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 $_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_{f}$ 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 $_{f}$ (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.
LA - eng
KW - rational points; châtelet threefold; unirationality; norm form; cubic extension
UR - http://eudml.org/doc/279029
ER -