# Arithmetic and Algebraic Geometry Seminar/3rd Year Candidacy Requirement: "K-theoretic criteria for exactness of some families of Laurent Polynomials"

*Abstract: Let P be an irreducible 2-variable Laurent polynomial, and let C be the curve it cuts out in (C*)^2. We can define a 1-form eta on (C*)^2 such that its restriction is a closed 1-form on C. We say P is exact if this restriction is exact on a smooth projective model of C, and we let V denote the primitive.*

*To any hyperbolic 3-manifold M with 1-torus boundary component one can define a polynomial invariant called its A-polynomial. It's known that factors of A-polynomials are exact: if we consider the symbol {x,y} in Milnor K_2 of the curve C, its defining polynomial P is an A-factor iff the symbol is torsion (which implies exactness of eta on the normalization). If M admits a complete hyperbolic structure there is a point in the curve C defined by its A-polynomial such that V(x,y) = Vol(M). *

*Now take a Newton polygon with g interior integer points. Consider the g-parameter family of Laurent polynomials P obtained by varying the corresponding coefficients while fixing the edge polynomials (so that their roots are roots of unity). Then we expect only finitely many of these P (in this family) are exact. This is closely related to the behavior of the logarithmic Mahler measure of P, and questions about the arithmetic of degenerations in the case M is an arithmetic 3-manifold.*

*Host: Matt Kerr*