On the local solubility of diophantine systems

TD Wooley*

*Corresponding author for this work

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

23 Citations (Scopus)


Let p be a rational prime number. We refine Brauer's elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (rd(2))(2d-1). This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q(p), and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.

Original languageEnglish
Pages (from-to)149-165
Number of pages17
JournalCompositio Mathematica
Issue number2
Publication statusPublished - Apr 1998


  • local solubility
  • diophantine equations
  • forms in many variables
  • p-adic fields


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