A Pair of Additive Quartic Forms

Hemar Godinho

Davenport and Lewis in a series of papers (see [12], [13], [14]) pioneered the study of systems of 2 , and later R > 2, additive forms, determining sufficient conditions for these systems to have rational zeros (i.e., non-trivial solutions). In the cas e of systems of two additive forms of the same low odd degree definite improvements on their results have been made. Their p-adic results, for the case of quintics, were improved by Cook[7, 8], and their results on integral zeros for a pair of cubic forms (see [12]) were improved by Vaughan[20] who obtained the best possible result. For pairs of cubics, assuming an extra hypothesis, Cook[6] obtained a more precise p-adic result, and Baker and Brüdern[2] , a more precise result for rational zeros than that of Vaughan.

In this thesis we discuss the case of rational solutions for a pair of additive quartic forms with rational coefficients. Essential to our work is the study of p-adic solutions, and we shall 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 :

1. If n ≥ 17 then f, g have a zero in all p-adic fields for p ≥ 677;

2. If n ≥ 21 then f, g have a zero in all p-adic fields for p ≥ 61, except possibly p = 73;

3. If n ≥ 25 then f, g have a zero in all p-adic fields for p ≠ 2, 5;

4. If n ≥ 33 then f, g have a zero in all p-adic fields for p ≠ 2;

5. If n ≥ 61 then f, g have a zero in all p-adic fields .

We deduce by means of the Hardy-Littlewood Method, the following theorem Theorem 2: Let f, g be a pair of additive quartic forms with integral coefficients, and assume that the following conditions are satisfied

1. any form h = λf + μg, where λ, μ are real numbers not both zero, has at least 17 variables explicitly;

2. f, g have a non-singular real zero and a non-singular 2-adic zero;

3. n ≥ 33;

then f, g have infinitely many integer zeros.

For the case of quartic forms our p-adic result is a definite improvement over the more general result obtained by Davenport and Lewis in [13]. Our p-adic results when p is odd, are obtained by use of exponential sums and computation in the case of small primes. The major difficulty in our proof lies in demonstrating the existence of 2-adic zeros, and that analysis constitutes the bulk of this thesis. In our treatment of the 2-adic case we extend the ideas of Davenport and Lewis on contraction of variables as exibited in [13], and the argument is very combinatorial in nature.

We do not know if 61 is the minimum number of variables needed for 2-adic solutions, but we have some indication that we need more than 57 variables, and the question remains open. But we do show in the course of our discussion that the lower bounds on the number of variables are sharp in cases 1,3,4 of Theorem 1.