Forms in Many Variables over Local Fields

University Of Rochester, Rochester NY

Investigators

Abstract

The investigator studies conditions under which systems of homogeneous polynomials in many variables have nontrivial zeros in p-adic completions of the rational numbers. A conjecture usually attributed to Artin states that if the coefficients of the polynomials are all integers, then such zeros exist in all p-adic completions provided that the number of variables is large enough when compared to the degrees of the polynomials, and gives a bound on the required number of variables. While the first part of this conjecture has been shown to be correct, the conjectured bound is false in general. However, the bound is correct if the prime p is sufficiently large. This leads to the questions of what the correct general bound should be and also how large the prime p needs to be for Artin's bound to hold. The investigator studies these questions in general, and also when the polynomials are restricted to being additive (i.e. having no cross-terms). These questions are studied both when the degrees of the polynomials are all equal and when they may vary, and are also studied in the more general setting where the p-adic completions of the rational numbers are replaced by extensions of these fields. Since the time of the ancient Greeks, a fundamental question in number theory has been to find solutions of (systems of) equations in which all of the variables are integers. When attempting to show that such solutions exist, it is often first necessary to obtain information about the so-called "local" solutions. These are solutions in the number systems of "p-adic integers," where p may be any prime number. In fact, it turns out that if there is even one prime p for which the system does not have a solution in p-adic integers, then it has no solutions in ordinary integers either. There are also some special situations in which the converse holds: if the system has solutions in p-adic integers for every prime p, then it automatically has a solution in regular integers. Hence it is an interesting problem to attempt to determine when equations have p-adic solutions. Such questions are also interesting apart from their connection to integral solutions of equations, since they provide insight into the nature of p-adic numbers, and how they differ from ordinary real numbers.