Representing integers by multilinear polynomials (Lenny Fukshansky, CMC)

Given a homogeneous multilinear polynomial F(x) in n variables with integer coefficients, we obtain some sufficient conditions for it to represent all the integers. Further, we derive effective results, establishing bounds on the size of a solution x to the equation F(x) = b, where b is any integer. For a special class of polynomials coming from determinants of rectangular matrices we are able to obtain necessary and sufficient conditions for such an effective representation problem. This result naturally connects to the problem of extending a collection of primitive vectors to a basis in a lattice, where we present counting estimates on the number of such extensions. Equivalently, this can be described as the number of ways a rectangular integer matrix can be extended to a matrix in GL_n(Z), when such extensions are possible. The talk is based on joint works with A. Boettcher and with M. Forst.