RedDots: Magnetic field of the nearby active M dwarf GJ 729, and a search for companions
M dwarfs are prime targets for discovering exoplanets, and the nearest M dwarfs to the Sun provide among the best opportunities for follow-up detailed exoplanet characterization. GJ 729, the seventh closest M dwarf to the Sun, presents significant challenges for exoplanet detection due to its high levels of magnetic activity. To address this, we present a detailed analysis of GJ 729's magnetic field and its variability, followed by a search for exoplanets beneath the activity-induced noise in the stellar radial velocity. The geometry of GJ 729's large-scale magnetic field was reconstructed using new and archival spectropolarimetric data for a total of four epochs spanning 10 years. Results indicate a weak large-scale field ranging from 50 to 145 G, and an evolving non-axisymmetric field geometry that varies from poloidal dominated to a near-equal poloidal-toroidal configuration. We modeled activity-induced radial velocity variations using Gaussian Process Regression and activity diagnostics, and searched for planetary companions using ~90 d of high-cadence spectra taken contemporaneously with the high-precision CARMENES and HARPS spectrographs. Activity-only and activity + Keplerian models offered statistically equivalent fits, with a consistently preferred Keplerian period of ~7 d and amplitude of ~1.9 m/s across a range of activity modeling approaches. This could relate to an Earth-mass or Super-Earth planet, or residual stellar activity with power concentrated at a multiple of the rotation half-period. Our findings provide insight into the magnetic behavior of fully convective M dwarfs, and highlight the potential and challenges of detecting Keplerian RV signatures that are only a fraction of activity amplitudes.