State-of-the-art approaches to magnetic anomaly detection rely on the generalized likelihood ratio test (GLRT). These approaches are based on the formulation of a parametric model of the source to be detected, expressed in a suitable functional basis. One of the primary objectives of this study is to demonstrate that, for a given measurement configuration, the signal is constrained to evolve within a restricted subset of the space generated by these functional bases. The parametric representation of the signal is identified as a semi-algebraic space which, for the dipole model used in this article, turns out to be a cone outside of which the estimated signal does not satisfy the physical equations. Thus, a second objective is to exploit this property to constrain the signal parameters in the GLRT to belong to the semi-algebraic space, in order to improve detection performance. The performance gain of the proposed algorithm is compared to the one of conventional approaches; numerical simulations show that the proposed approach not only outperforms state-of-the-art methods but can even provide results close to those of the clear-seeing (optimal) receiver.