We verify a case of Artin's conjecture for a certain range by showing that every cubic and quadratic form over a p-adic field with at least 14 variables has a non-trivial common zero, provided the cardinality of the residue class field exceeds 293. The proof involves generalizing a p-adic minimization procedure due to Schmidt, to hold for systems of forms of differing degrees.
We show that all p-adic quintic forms in at least n > 4562911 variables have a non-trivial zero. We also derive a new result concerning systems of cubic and quadratic forms.
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