The theory of the sign detection method that computes the signs of arbitrary integer homogeneous polynomials is presented. The robustness of geometric algorithms is an important issue in geometric modeling. One answer to the robustness is to employ the exact integer arithmetic. This approach makes it possible to achieve the complete robustness of geometric algorithms. In geometric algorithms, signs of polynomials, such as determinants or inner products, are necessary in most cases. The sign detection method determines the signs of polynomials without evaluating them exactly, and improves the computational cost of the exact integer arithmetic. By homogenizing Euclidean coordinates, hence by adding the fourth coordinate in case of 3-dimensional Euclidean coordinates, homogeneous coordinates are obtained. Polynomials that are non-homogeneous become homogeneous by the homogenization. A generalized theory for the sign detection of arbitrary homogeneous polynomials and some characteristics of it are presented.