Improvements in Birch's theorem on forms in many variables

Tim D Browning, Sean M Prendiville

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

8 Citations (Scopus)
342 Downloads (Pure)


We show that a non-singular integral form of degree d is soluble over the integers if and only if it is soluble over ℝ and over ℚp for all primes p, provided that the form has at least (d - d1/2/2)2d variables. This improves on a longstanding result of Birch.
Original languageEnglish
Pages (from-to)203-234
Number of pages32
JournalJournal für die reine und angewandte Mathematik
Issue number731
Early online date20 Feb 2015
Publication statusPublished - 1 Oct 2017


  • math.NT
  • 11P55 (11G35, 14G05)


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