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We prove two conjectures from [M. R. Douglas, B. Shiffman and S. Zelditch,Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics.J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the expected numberof critical points of random holomorphic sections of a positive line bundle. Weshow that, on average, the critical points of minimal Morse index are the mostplentiful for holomorphic sections of ${\mathcal O}(N) \to \CP^m$ and, in anasymptotic sense, for those of line bundles over general K\"ahler manifolds. Wecalculate the expected number of these critical points for the respective casesand use these to obtain growth rates and asymptotic bounds for the totalexpected number of critical points in these cases. This line of research wasmotivated by landscape problems in string theory and spin glasses.Comment: 14 pages, corrected typo

Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections