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Hemar Godhino; Paulo H. A. Rodrigues
On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 205-219, doi: 10.5802/jtnb.582
Article PDF | Reviews MR 2332062 | Zbl 1131.11023

Résumé - Abstract

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $ax^k+by^k+cz^k \equiv d \,(\mbox {mod}\,p)$ has a solution with $xyz \nequiv0 \,(\mbox {mod}\,p)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2\cdot 3^{R-1}\cdot k +1$ variables, has always non-trivial $p$-adic solutions, provided $p \nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p > k^4$.

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