Ricci, J. L. (2014). Finiteness Results for Regular Ternary Quadratic Polynomials. Retrieved from https://doi.org/10.14418/wes01.3.33
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, 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. In this talk, I will give a 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 will 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.