We show that a sum of four non-degenerate binary cubic forms with integral coefficients necessarily possesses a non-trivial rational zero. When each of these binary cubic forms has non-zero discriminant, we are able to obtain bounds on the number, N(P), of integral zeros of the sum inside a box of size P of the shape
P5-epsilon much less than(epsilon) N(P) much less than(epsilon) P5+epsilon.
Finally, given two binary cubic forms with non-zero discriminant, we show that almost all integers, lying in those congruence classes permitted by local solubility conditions, are represented as the sum of the aforementioned forms.
|Number of pages||37|
|Journal||Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|
|Publication status||Published - 15 Mar 1998|
- Hardy-Littlewood method
- exponential sums
- diophantine equations
- cubic forms
- representation problems
- efficient differencing
- WARING PROBLEM