Simultaneous zeros of a Cubic and Quadratic form

Details Simultaneous zeros of a Cubic and Quadratic form Simultaneous zeros of a Cubic and Quadratic form

2010-01-07

arxiv.org/abs/1001.1055v1

Mathematics

Number Theory

19 pages

Scientific paper

We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic form over any $p$-adic field, provided the cardinality of the residue class field exceeds 293. That is any Cubic and Quadratic form with at least 14 variables has a non-trivial $p$-adic zero, with the aforementioned condition on the residue class field. A crucial step in the proof, involves generalizing a $p$-adic minimization procedure due to W. M. Schmidt to hold for systems of forms of arbitrary degrees.

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