Simultaneous zeros of a Cubic and Quadratic form

TL;DR

This paper proves a case of Artin's conjecture for systems of a cubic and quadratic form over p-adic fields with large residue fields, establishing the existence of non-trivial zeros under certain conditions.

Contribution

It generalizes a p-adic minimization method to systems of forms of arbitrary degrees and verifies Artin's conjecture for a specific case involving cubic and quadratic forms.

Findings

Any cubic and quadratic form with at least 14 variables over a p-adic field has a non-trivial zero if the residue field size exceeds 293.

The p-adic minimization procedure is extended to systems of forms of arbitrary degrees.

The conjecture holds under the specified conditions on the residue class field size.

Abstract

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.