PUB - Publikationen an der Universität Bielefeld

Asymptotic expansions for the number of lattice points in regions of the d-dimensional Euclidean space are obtained. We assume that the regions are level sets of polynomials of degree p greater than or equal to 2. The expansions are valid provided that some 'local' bounds on trigonometric sums related to these polynomials are satisfied. Using this approach we obtain as an initial application optimal explicit bounds for the lattice point rest in large bounded bodies generated by polynomials slightly more general than Q(x) = Q(p)(x) + P(x), where p is an even number, the polynomial is given by Q(p)(x) = lambda (l)x(l)(p) + ... + lambda (d)x(d)(p), for x = (x(l),...,x(d)) is an element of R-d, with some lambda (j) > 0, and P is a polynomial of degree strictly less than p. The bounds are optimal and hold for sufficiently large dimensions d greater than or equal to c(p). The proofs are based on extensions of techniques developed for quadratic forms by the authors. Similar to the quadratic case, the results yield variants of the quantitative Oppenheim and Davenport-Lewis conjectures for classes of higher order polynomials in sufficiently large dimensions.