Computing integral points in convex semi-algebraic sets

Let Y be a convex set in R^k defined by polynomial inequalities and equations of degree at most d⩾2 with integer coefficients of binary length l. We show that if Y∩Z^k≠θ, then Y contains an integral point of binary length ld^O((k⁴)). For fixed k, our bound implies a polynomial-time algorithm for computing an integral point y∈Y. In particular, we extend Lenstra´s theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming