Marino, A. (2017). Finiteness of Strictly n-Regular Quadratic Forms. Retrieved from https://doi.org/10.14418/wes01.3.70

A quadratic form is a homogeneous polynomial of degree two. In the arithmetic theory of integral quadratic forms, a main question is the representation problem: given an integral quadratic form f, for which integers a does there exist a solution to f(x) = a? We call an integral quadratic form regular if the existence of solutions locally everywhere implies the existence of a solution over the rational integers. We can strengthen this notion of regularity to strict regularity by demanding that the solutions are primitive, i.e. the coordinates of the solutions are coprime. In 2014, Earnest-Kim-Meyer proved that there are finitely many equivalence classes of primitive positive definite integral strictly regular quadratic forms in four variables. The main result of this thesis extends their result in the context of a higher dimensional analogue of strict regularity. We obtain the following result: for n >= 2, there are only finitely many equivalence classes of primitive positive definite integral strictly n-regular quadratic forms of n + 4 variables.