Quantifier Elimination in p-adic Fields

We present a tutorial survey of quantifier elimination and decision procedures in p-adic fields. The p-adic fields are studied in the (so-called) P n -formalism of Angus Macintyre, for which motivation in provided through a rich body of analogies with real-cloned fields. Quantifier elimination and decidion procedures are described proceeding via a Cylindrical Algebraic Decomposition of affine p-adic space. effective complexity analyses are also provided

[1]  John H. Reif,et al.  The complexity of elementary algebra and geometry , 1984, STOC '84.

[2]  Jean-Pierre Serre.  A Course in Arithmetic , 1973 .

[3]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[4]  S. Kochen,et al.  Diophantine Problems Over Local Fields I , 1965 .

[5]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[6]  Martin Fürer,et al.  The Complexity of Presburger Arithmetic with Bounded Quantifier Alternation Depth , 1982, Theor. Comput. Sci..

[7]  Angus Macintyre,et al.  On definable subsets of p-adic fields , 1976, Journal of Symbolic Logic.

[8]  M. Fischer,et al.  SUPER-EXPONENTIAL COMPLEXITY OF PRESBURGER ARITHMETIC , 1974 .

[9]  M. A. Dickmann,et al.  Applications of model theory to real algebraic geometry , 1985 .

[10]  Jan Denef,et al.  p-adic semi-algebraic sets and cell decomposition. , 1986 .

[11]  Volker Weispfenning,et al.  Quantifier elimination and decision procedures for valued fields , 1984 .

[12]  Paul J. Cohen,et al.  Decision procedures for real and p‐adic fields , 1969 .

[13]  Jan Denef,et al.  The rationality of the Poincaré series associated to thep-adic points on a variety , 1984 .

[14]  L. Dries.  Some applications of a model theoretic fact to (semi-) algebraic geometry , 1982 .

[15]  Irving Kaplansky,et al.  Hilbert's problems , 1977 .

[16]  Volker Weispfenning,et al.  On the elementary theory of Hensel fields , 1976 .

[17]  A. Prestel,et al.  Lectures on formally p-adic fields , 1983 .