# zbMATH — the first resource for mathematics

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
Full Text: