Additive forms of degree \(2^ l\)
Let \(a_ 1, \dots, a_ n\); \(b_ 1, \dots, b_ n\) be rational and \(k\) be a positive integer. \textit{H. M. Davenport} and \textit{D. J. Lewis} [Proc. Symp. Pure Math. 12 (1967), 74-98 (1969; Zbl 0226.10026)] proved that if \(n \geq 7 k^ 3\) then the pair of additive forms \[ f=a_ 1x_ 1^ k + \cdots + a_ nx_ n^ k, \quad g = b_ 1x_ 1^ k + \cdots + b_ nx_ n^ k \] has \(p\)-adic zeros for every prime \(p\). For odd \(k\) they obtained a better condition, namely, \(n \geq 2k^ 2 + 1\). In this paper, the author deals with the special case for even \(k\) and proves that if \(k = 2^ l\), \(l \geq 2\) and if \(n \geq 16k^ 2 - 26k + 1\) then \(f,g\) have common \(p\)-adic zeros for all primes \(p\). The techniques used in the proof are an extension of those in the author's Ph.D. thesis [A pair of additive quartic forms, Univ. Michigan, 1992].
scientific article; zbMATH DE number 584055