Quintic forms over \(p\)-adic fields
Let \(K\) be a complete, discretely valued field with finite residue class field \(k\). We say that \(K\) has the property \(C_2 (d)\) if every homogeneous form defined over \(K\) of degree \(d\) in greater than \(d^2\) variables has a nontrivial zero. In this paper, with the help of results in \textit{R. R. Laxton} and \textit{D. J. Lewis} [Proc. Symp. Pure Math. 8, 16-21 (1965; Zbl 0277.10018)], the authors obtain: Let \(K\) be any \(p\)-adic field with residue class field \(k\) of cardinality \(q\geq 47\). Then \(K\) has the property \(C_2 (5)\). It was mentioned by the authors as a footnote in the paper that \textit{J.-P. Serre} has informed the authors that he can lower the bound in the above result to \(q\geq 43\) by extending his methods in his papers [Coll. Papers III, 664-668 (1986) = Sémin. Théor. Nombres Bordeaux 1982/83, No. 22 (1983; Zbl 0538.14016) and Coll. Papers III, 701-705 = Annuaire du Collège de France 1983-84, 79-83 (1984)].
scientific article; zbMATH DE number 874194