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We investigate the ability of three-integral, axisymmetric, orbit-basedmodeling algorithms to recover the parameters defining the gravitationalpotential (M/L ratio and black hole mass Mh) in spheroidal stellar systemsusing stellar kinematical data. We show that the potential estimation problemis generically under-determined when applied to long-slit kinematical data ofthe kind used in most black hole mass determinations to date. A range ofparameters (M/L, Mh) can provide equally good fits to the data, making itimpossible to assign best-fit values. We illustrate the indeterminacy using avariety of data sets derived from realistic models as well as publishedobservations of the galaxy M32. In the case of M32, our reanalysis demonstratesthat data published prior to 2000 are equally consistent with Mh in the range1.5x10^6-5x10^6 solar masses, with no preferred value in that range. While theHST/STIS data for this galaxy may overcome the degeneracy in Mh, HST data formost galaxies do not resolve the black hole's sphere of influence and in thesegalaxies the degree of degeneracy allowed by the data may be substantial. Weinvestigate the effect on the degeneracy of enforcing smoothness(regularization) constraints. However we find no indication that the truepotential can be recovered simply by enforcing smoothness. For a givensmoothing level, all solutions in the minimum-chisquare valley exhibit similarlevels of noise. These experiments affirm that the indeterminacy is real andnot an artifact associated with non-smooth solutions. (Abridged)Comment: Accepted for publication in The Astrophysical Journal. Changes  include discussion of regularizatio

Difficulties with Recovering the Masses of Supermassive Black Holes from Stellar Kinematical Data