Context. The magnetic fields of the accreting white dwarfs in magnetic cataclysmic variables (mCVs) determine the accretion geometries, the emission properties, and the secular evolution of these objects.\ud Aims. We determine the structure of the surface magnetic fields of the white dwarf primaries in magnetic CVs using Zeeman tomography.\ud Methods. Our study is based on orbital-phase resolved optical flux and circular polarization spectra of the polars EF Eri, BL Hyi, and CP Tuc obtained with FORS1 at the ESO VLT. An evolutionary algorithm is used to synthesize best fits to these spectra from an\ud extensive database of pre-computed Zeeman spectra. The general approach has been described in previous papers of this series.\ud Results. The results achieved with simple geometries as centered or offset dipoles are not satisfactory. Significantly improved fits are obtained for multipole expansions that are truncated at degree lmax = 3 or 5 and include all tesseral and sectoral components with\ud 0 ≤ m ≤ l. The most frequent field strengths of 13, 18, and 10MG for EF Eri, BL Hyi, and CP Tuc, and the ranges of field strength covered are similar for the dipole and multipole models, but only the latter provide access to accreting matter at the right locations on the white dwarf. The results suggest that the field geometries of the white dwarfs in short-period mCVs are quite complex, with\ud strong contributions from multipoles higher than the dipole in spite of a typical age of the white dwarfs in CVs in excess of 1 Gyr.\ud Conclusions. It is feasible to derive the surface field structure of an accreting white dwarf from phase-resolved low-state circular spectropolarimetry of sufficiently high signal-to-noise ratio. The fact that independent information is available on the strength and\ud direction of the field in the accretion spot from high-state observations helps in unraveling the global field structure
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