Computing integral points in convex semi-algebraic sets

Let Y be a convex set in R/sup k/ defined by polynomial inequalities and equations of degree at most d/spl ges/2 with integer coefficients of binary length l. We show that if Y/spl cap/Z/sup k//spl ne//spl theta/, then Y contains an integral point of binary length ld/sup O/((k/sup 4/)). For fixed k, our bound implies a polynomial-time algorithm for computing an integral point y/spl isin/Y. In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming.