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We study the integer minimization of a quasiconvex polynomial withquasiconvex polynomial constraints. We propose a new algorithm that is animprovement upon the best known algorithm due to Heinz (Journal of Complexity,2005). This improvement is achieved by applying a new modern Lenstra-typealgorithm, finding optimal ellipsoid roundings, and considering sparseencodings of polynomials. For the bounded case, our algorithm attains atime-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a boundon the number of monomials in each polynomial and r is the binary encodinglength of a bound on the feasible region. In the general case, s l^{O(1)}d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the totaldegree of the polynomials and l bounds the maximum binary encoding size of theinput.

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)