On Almost Universal Ternary Inhomogeneous Quadratic Polynomials

A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A related and equally interesting problem is the representation of integers by inhomogeneous quadratic polynomials. An inhomogeneous quadratic polynomial is a sum of a quadratic form and a linear form; it is called almost universal if it represents all but finitely many positive integers. This thesis gives a characterization of almost universal ternary inhomogeneous quadratic polynomials, H(x) whose quadratic parts are positive definite and anisotropic at exactly one prime. Imposing some other mild arithmetic conditions, we utilize the theory of quadratic lattices and primitive spinor exceptions to give a list of explicit conditions, under which H(x) is almost universal. In the final chapter, we will give some examples of almost universal quadratic polynomials given by mixed sums of polygonal numbers.