Finiteness Results for Regular Ternary Quadratic Polynomials

Abstract

In 1924, Helmut Hasse established a local-to-global principle for representations of rational quadratic forms. Unfortunately, an analogous local-to-global principle does not hold for representations over the integers. A quadratic polynomial is called regular if such a principle exists; that is, if it represents all the integers which are represented locally by the polynomial itself over Zp for all primes p as well as over R. In 1953/54 [16, 17], G.L. Watson showed that up to equivalence, there are only finitely many primitive positive definite integral regular quadratic forms in three variables. More recently, W.K. Chan and B.-K. Oh take the first step in understanding regular ternary quadratic polynomials by showing that there are only finitely many primitive positive regular triangular forms in three variables [6]. This thesis gives an analogous finiteness result for regular ternary quadratic polynomials in greater generality. By defining an invariant called the conductor and a notion of a semi-equivalence class of a quadratic polynomial, we utilize the theory of quadratic forms to obtain the following result: Given a fixed conductor, there are only finitely many semi-equivalence classes of positive regular quadratic polynomials in three variables.