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Simultaneous diagonal \(p\)-adic equations. (English) Zbl 0992.11030

O. D. Atkinson, J. Brüdern and R. J. Cook [Mathematika 39, 1-9 (1992; Zbl 0774.11016)] proved the following result for systems of additive equations over the field of \(p\)-adic numbers:
Theorem: Let \(r\), \(k\), \(n\) be positive integers with \(k>1\) and \(n> 2rk\). Then the system of equations \[ F_i(x)= a_{i1}x_1^k+\cdots+ a_{in}x_n^k= 0, \quad i=1,\dots, r,\tag{1} \] with coefficients \(a_{ij}\in \mathbb{Z}\), has a nontrivial \(p\)-adic solution for all \(p>k^{2r+2}\).
In this paper the author considers the same problem for equations in \(n>crk\) variables where \(c\) is a positive integer and \(c\geq 3\). Using a similar approach as Atkinson et al., differing mainly in the exponential sum arguments and the handling of singular solutions, the author proves:
Theorem 1. Let \(r\), \(k\), \(n\), \(c\) be positive integers with \(k>1\), \(n> crk\) and \(c>2\). Then the system of equations (1), with coefficients \(a_{ij}\in \mathbb{Z}\), has a nontrivial \(p\)-adic solution for all \(p>r^2 k^{2+(2/(c-2))}\) if \(r\neq 1\) and \(p> k^{2+(2/(c-1))}\) if \(r=1\).
The \(p\)-adic solutions are obtained by a Hensel’s Lemma argument from nonsingular solutions to certain congruences. However, the congruences \(\pmod p\) are solved by an induction argument on \(r\). It turns out that not only homogeneous congruences, but also inhomogeneous congruences must be considered. For these the author derives an interesting result of independent interest as follows.
Let \(B= (b_{ij})\) be an \(r\times m\) matrix over the field \({\mathbf K}\). For \(0\leq d\leq r\) denote by \(\mu(d,B)\) the maximum number of columns from \(B\) that can lie in a \(d\)-dimensional subspace of \({\mathbf K}^r\). If \(m= cr+1\) and \(\mu(d,B)\leq cd\) holds for all \(d\leq r-1\) then \(B\) is called highly nonsingular. For \(b_{ij}, d_i\in \mathbb{Z}\) a solution \({\mathbf x}\) to the congruences \[ b_{i1}x_1^k+\cdots+ b_{im}x_m^k \equiv d_i\pmod p, \quad i=1,\dots, r, \tag{2} \] is said to be of rank \(\rho\) if the matrix \((b_{ij}x_j)\) has rank \(\rho\) in \(\mathbb{F}_p\), and is nonsingular if it has maximal rank.
Theorem 2. Let \(m= cr+1\), \(b_{ij},d_i\in \mathbb{Z}\) be such that \(B= (b_{ij})\) is highly nonsingular in the field \(\mathbb{F}_p\). Then the system of congruences (2), \(\operatorname {mod}p\), has a nonsingular solution \(\operatorname {mod}p\) for all primes \(p> r^2 k^{2+(2/(c-2))}\) if \(r\neq 1\) and \(p> k^{2+(2/(c-1))}\) if \(r=1\).
For the proof of Theorem 1 the author only requires Theorem 2 in the case \({\mathbf K}= \mathbb{F}_p\), however the result also holds more generally for finite fields \({\mathbf K}= \mathbb{F}_q\) \((q=p^f)\).

MSC:

11D88 \(p\)-adic and power series fields
11D72 Diophantine equations in many variables
11D79 Congruences in many variables

Citations:

Zbl 0774.11016
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References:

[1] Hardy, Inequalities (1952)
[2] DOI: 10.1098/rsta.1969.0035 · Zbl 0207.35304 · doi:10.1098/rsta.1969.0035
[3] Schmidt, Equations over Finite Fields, An Elementary Approach, Lecture Notes in Mathematics 536 (1976) · Zbl 0329.12001 · doi:10.1007/BFb0080437
[4] Aigner, Combinatorial Theory (1979) · doi:10.1007/978-1-4615-6666-3
[5] Atkinson, Mathematika 39 pp 1– (1992)
[6] Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities (1963) · Zbl 1089.11500
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