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On zeros of \(p\)-adic forms. (English) Zbl 0531.10026
Let \(F\) be a form of degree \(d\), in \(n\) variables and with coefficients in a \(p\)-adic field \({\mathbb{Q}}_ p\). Suppose that \(F\) has a non-trivial zero in \({\mathbb{Q}}_ p\) when \(n>\phi_ p(d)\). Artin conjectured that \(\phi_ p(d)=d^ 2\) but Terjanian constructed a form of degree 4 in 18 variables which did not represent \(0\) non-trivially in \({\mathbb{Q}}_ 2\) [G. Terjanian, C. R. Acad. Sci., Paris, Sér. A 262, 612 (1966; Zbl 0133.297)]. Browkin extended Terjanian’s construction to obtain counter-examples for each prime \(p\), but always with \(\phi_ p(d)<d^ 3\). [J. Browkin, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 489–492 (1966; Zbl 0139.282)]. These results have been greatly improved by G. I. Arkhipov and A. A. Karatsuba [Sov. Math., Dokl. 25, 1–3 (1982); translation from Dokl. Akad. Nauk SSSR 262, 11–13 (1982; Zbl 0496.10009)], who showed that for each prime \(p\) there are infinitely many \(d\) such that \(\phi_ p(d)>\exp(d/(\log d)^ 2(\log \log d)^ 3).\) In this paper the authors obtain a slightly better result by using a more efficient \(p\)-adic interpolation argument. They prove that given any prime \(p\) and any \(\epsilon>0\), there exists for infinitely many \(d\) a form \(F\) of degree \(d\) with \[ (*)\quad n>\exp(d/(\log d)(\log \log d)^{1+\epsilon}) \] such that if \(F(a_ 1,\dots,a_ n)\equiv 0 (mod p^ d)\), \(a_ 1,\dots,a_ n\in {\mathbb{Z}}\), then \(a_ 1\equiv\dots \equiv a_ n\equiv 0~(mod p)\). Results comparable to these have been established by W. D. Brownawell [J. Number Theory 18, 342–349 (1984; Zbl 0531.10027)].
It is not known how close to best possible the lower bound (*) is. Brauer showed that \(\phi_ p(d)\) was finite and his arguments were extended by D. B. Leep and W. M. Schmidt [Invent. Math. 71, 539–549 (1983; Zbl 0504.10010)] to give an upper bound \(C\quad \exp((d!)^ 2(1+\epsilon)^ d),\) where \(C>0\) is a constant, for \(\phi_ p(d)\). But by refining the arguments, W. M. Schmidt has now improved this estimate to \(\phi_ p(d)<\exp(2^ d d!)\) [J. Number Theory 19, 63–80 (1984; Zbl 0541.10024)]. This is still a long way from (*).
It is known that there exists a function \(p_ 0(d)\) such that \(p>p_ 0(d)\) implies \(\phi_ p(d)=d^ 2\), and the authors point out that in all the known examples where \(\phi_ p(d)>d^ 2\) and \(p>2\), \(d\) is divisible by \(p-1\). They suggest that these are the only exceptions to \(\phi_ p(d)=d^ 2\); and that \(\phi_ p(d)=d^ 2\) for all \(p\) only when \(d\) is a prime (by contrast with additive forms of degree \(d\) where \(d^ 2+1\) variables always suffices for non-trivial \(p\)-adic zeros).

MSC:
11E95 \(p\)-adic theory
11E76 Forms of degree higher than two
11D72 Diophantine equations in many variables
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