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On the local solubility of diophantine systems. (English) Zbl 0892.11011

For an \(r\)-tuple of polynomials \(f=(f_1, \dots, f_r)\) with coefficients in a field \(k\), let \(\nu(f)\) be the number of variables appearing explicitly in \(f\). By \({\mathcal H}_{d,r} (k)\) we denote the set of all \(r\)-tuples of homogeneous polynomials of degree \(d\) with coefficients in \(k\) which possess no nontrivial zeros over \(k\), and by \({\mathcal D}_{d,r} (k)\) we denote the corresponding set of all diagonal homogeneous polynomials. Put \(u_{d,r} (k)= \sup_{g\in {\mathcal H}_{d,r} (k)} \nu(g)\), \(\varphi_{d,r} (k)= \sup_{f\in {\mathcal D}_{d,r} (k)} \nu(f)\), and \(\varphi_d (k)= \varphi_{d,1} (k)\). In this paper the author obtains that if \(\varphi_i (k)< \infty\) for \(2\leq i\leq d\), then \[ u_{d,r} (k)\leq 2r^{2^{d-1}} \varphi_d^{2^{d-2}} \prod^{d-1}_{i=2} (\varphi_i+ 1)^{2^{i-2}}. \] In a practical sense, the result essentially improves on a previous known result by D. B. Leep and W. M. Schmidt [Invent. Math. 71, 539-549 (1983; Zbl 0504.10010)]. The author also obtains the following upper bounds for \(u_{d,r} (k)\) where \(k\) is of several fields of interest: \[ u_{d,r} (\mathbb{Q}_p) \leq(rd^2)^{2^{d-1}},\;u_{d,r} (K)\leq r^{2^{d-1}} e^{2^{d+2}} (\log d)^2, \]
\[ u_{d, r} (L)\leq r^{2^{d-1}} e^{2^dd},\;u_{d,r} (\mathbb{Q}^{\text{rad}}) \leq(2r^2)^{2^{d-2}}, \] where \(K\) is a finite extension of \(\mathbb{Q}_p\), \(L\) is a purely imaginary field extension of \(\mathbb{Q}\) and \(\mathbb{Q}^{\text{rad}}\) is the radical closure of \(\mathbb{Q}\).

MSC:

11D72 Diophantine equations in many variables
14G20 Local ground fields in algebraic geometry
11G25 Varieties over finite and local fields
11E95 \(p\)-adic theory
11E76 Forms of degree higher than two

Citations:

Zbl 0504.10010
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