Meir, I. D. Simultaneous diagonal \(p\)-adic equations. (English) Zbl 0992.11030 Mathematika 45, No. 2, 337-349 (1998). 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 \({\mathbf K}= \mathbb{F}_q\) \((q=p^f)\). Reviewer: O.Ninnemann (Berlin) Cited in 1 Document MathOverflow Questions: Solutions to system of polynomial equations over finite fields MSC: 11D88 \(p\)-adic and power series fields 11D72 Diophantine equations in many variables 11D79 Congruences in many variables Keywords:systems of additive equations; nontrivial \(p\)-adic solution; inhomogeneous congruences Citations:Zbl 0774.11016 PDFBibTeX XMLCite \textit{I. D. Meir}, Mathematika 45, No. 2, 337--349 (1998; Zbl 0992.11030) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.