The addition of binary cubic forms

J Brudern*, TD Wooley

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)701-737
Number of pages37
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume356
Issue number1738
Publication statusPublished - 15 Mar 1998

Keywords

  • Hardy-Littlewood method
  • exponential sums
  • diophantine equations
  • cubic forms
  • representation problems
  • efficient differencing
  • WARING PROBLEM
  • CUBES

Cite this