CircleZ: Reliable Photometric redshifts for AGN computed using only photometry from Legacy Survey Imaging for DESI
(abridged)Photometric redshifts for AGN (galaxies hosting an accreting supermassive black hole in their center) are notoriously challenging and currently better computed via SED fitting, assuming that deep photometry for many wavelengths is available. However, for AGN detected all-sky, the photometry is limited and provided by different projects. This makes the task of homogenising the data challenging and is a dramatic drawback for the millions of AGN that wide surveys like SRG/eROSITA will detect. This work aims to compute reliable photometric redshifts for X-ray-detected AGN using only one dataset that covers a large area: the 10th Data Release of the Imaging Legacy Survey (LS10) for DESI. LS10 provides deep grizW1-W4 forced photometry within various apertures, thus avoids issues related to the cross-calibration of surveys. We present the results from CircleZ, a machine-learning algorithm based on a Fully Connected Neural Network. CircleZ uses training sample of 14,000 X-ray-detected AGN and utilizes multi-aperture photometry. The accuracy and the fraction of outliers reached in a test sample of 2913 AGN are 0.067 and 11.6%, respectively. The results are comparable to or better than those obtained previously for the same field but with much less effort. We further tested the stability of the results by computing the photometric redshifts for the sources detected in CSC2 and Chandra-COSMOS Legacy, reaching comparable accuracy as in eFEDS when limiting the magnitude of the counterparts with respect to the depth of LS10. The method applies to fainter samples of AGN using deeper optical data from future surveys (e.g., LSST, Euclid), granted LS10-like information on the light distribution beyond a morphological type is provided. With the paper, we release an updated version of the photometric redshifts (including errors and probability distribution function) for eROSITA/eFEDS.