Since the poles are the opposite (magnetically speaking) ends of a magnet, and the magnetic force is the strength a magnet exerts over bodies around, him, I'd say that the region of influence is the magnetic field domain.
The region of influence around a magnet is called the magnetic field.
Explanation;
Magnetic field is the space all around a magnet where the force of the magnet can act . It is the region surrounding a magnet which exhibits a magnetic force.
Magnetic field lines map out the magnetic field around a magnet. The magnetic field of a bar magnet is strongest at either pole of the magnet.
Magnetic domain is a cluster of atoms that all have magnetic fields that are lined up in the same way.
