#### Publication Date

5-2014

#### Advisor(s)

Wai Kiu K. Chan

#### Department

Mathematics

#### Language

English

#### 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 ℤ* _{p}* 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.

#### Recommended Citation

Ricci, James Lewis, "Finiteness Results for Regular Ternary Quadratic Polynomials" (2014). *Dissertations*. 40.

http://wesscholar.wesleyan.edu/etd_diss/40

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