Bridging the algebraic and analytic theory of the representation problem for quadratic polynomials

Given a polynomial f(x) of several variables with rational coefficients and an integer n, we say that f represents n if the equation f(x)=n is solvable in the integers.  One might ask, is it possible to effectively determine the set of integers represented by f? This so-called representation problem for quadratic polynomials is one of the classical problems in number theory.  The negative answer to Hilbert's 10th problem tells us that in general, there is no finite algorithm to decide whether a solution exists.  However work of Siegel in the 1970’s shows that in the case of quadratic polynomials the representation problem is tractable.  Algebraically, we can view the representation problem using the language of integral quadratic lattices and lattice cosets, but this method stops short of realizing a full local-global principle.  Melding this with the analytic approach of studying the coefficients of theta series we can reach some satisfying solutions to the representation problem for certain families of polynomials.  In this talk we will illuminate several important connections between the algebraic and analytic theory of quadratic lattices and finish with a conjecture involving a Siegel-Well type formula for representations by inhomogeneous quadratic polynomials.