Solubility of additive forms of twice odd degree over ramified quadratic extensions of
Tom 201 / 2021
Acta Arithmetica 201 (2021), 149-164
MSC: 11D72, 11D88, 11E76.
DOI: 10.4064/aa201116-30-4
Opublikowany online: 28 October 2021
Streszczenie
We determine the minimal number \Gamma ^*(d, K) of variables which guarantees a nontrivial solution for every additive form of degree d=2m, m odd, m \ge 3 over the six ramified quadratic extensions of \mathbb {Q}_2. We prove that if K is one of \{\mathbb {Q}_2(\sqrt {2}), \mathbb {Q}_2(\sqrt {10}), \mathbb {Q}_2(\sqrt {-2}), \mathbb {Q}_2(\sqrt {-10})\}, then \Gamma ^*(d,K) = \frac {3}{2}d, and if K is one of \{\mathbb {Q}_2(\sqrt {-1}), \mathbb {Q}_2(\sqrt {-5})\}, \Gamma ^*(d,K) = d+1. The case d=6 was previously known.
