QUADRATIC-FORMS IN NORMAL OPEN INDUCTION

Models of normal open induction (NOI) are those discretely ordered rin gs, integrally closed in their fraction field whose nonnegative part s atisfy Peano's induction axioms for open formulas in the language of o rdered semirings. Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + bY2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a mo del of NOI. We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Th en we study the group of solutions of a Pell equation in NOI; we const ruct a model in which the quotient group by the squares has size conti nuum.