, Volume 93, Issue 3, pp 211-223

On cubic polynomials III. Systems ofp-adic equations

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Abstract

We study systems of equations $$\mathfrak{F}_1 = \ldots = \mathfrak{F}_r = 0,$$ where \(\mathfrak{F}_1 , \ldots ,\mathfrak{F}_r \) are cubic forms withp-adic coefficients. Such a system has a nontrivialp-adic solution if the number of variables is at least 50,000 ·r 3. Further we will give estimates for the number of solutions of certain systems of cubic congruences.

Partially supported by NSF contract NSF-MCS-8015356.