A pair of additive quartic forms.

Abstract

In this thesis we discuss the case of p-adic and rational zeros for a pair of additive quartic forms with rational coefficients, and we prove. Theorem 1. Let f,g be a pair of additive quartic forms with rational coefficients in n variables. Then the following statements are true: (a) If $n\ge 17$ then f,g have a zero in all p-adic fields for $p\ge 677$; (b) If $n\ge 21$ then f,g have a zero in all p-adic fields for $p\ge 61$, except p = 73; (c) If $n\ge 25$ then f,g have a zero in all p-adic fields for $p\not= 2,5$; (d) If $n\ge 33$ then f,g have a zero in all p-adic fields for $p\not= 2$; (e) If $n \ge 61$ then f,g have a zero in all p-adic fields. Theorem 2. Let f,g be a pair of additive quartic forms with integral coefficients, and assume that the following conditions are satisfied: (a) any form h = $\lambda f+\mu g$, where $\lambda,\mu$ are real numbers not both zero, has at least 17 variables; (b) f,g have a non-singular real zero and a non-singular 2-adic zero; (c) $n\ge 33$; then f,g have infinitely many integer zeros. The major difficulty in proving Theorem 1 is in showing the existence of 2-adic zeros, which is done by combinatoric type arguments. Our proof of Theorem 2 is by means of the Hardy-Littlewood Method, and essential to this proof is the study of p-adic solutions.