The paper by H. W. Lenstra, Jr. presents a polynomial algorithm for solving integer linear programming problems with a fixed number of variables. The main result shows that such problems can be solved in polynomial time, using methods from the geometry of numbers. The algorithm transforms the original problem into an equivalent one where either the existence of a solution is obvious, or the last coordinate of any solution belongs to a bounded interval. This reduction is achieved through two auxiliary algorithms: one that remodels the convex set and another that reduces the dimension of the lattice. The paper also discusses the case of a fixed number of constraints and the mixed integer linear programming problem, showing that these problems are also polynomially solvable. The algorithms are designed for theoretical purposes and may not be practical for large values of \( n \).The paper by H. W. Lenstra, Jr. presents a polynomial algorithm for solving integer linear programming problems with a fixed number of variables. The main result shows that such problems can be solved in polynomial time, using methods from the geometry of numbers. The algorithm transforms the original problem into an equivalent one where either the existence of a solution is obvious, or the last coordinate of any solution belongs to a bounded interval. This reduction is achieved through two auxiliary algorithms: one that remodels the convex set and another that reduces the dimension of the lattice. The paper also discusses the case of a fixed number of constraints and the mixed integer linear programming problem, showing that these problems are also polynomially solvable. The algorithms are designed for theoretical purposes and may not be practical for large values of \( n \).