Number of solutions of the TAP equations for p--spin interaction spin glasses

The number $\langle N_s\rangle$ of solutions of the equations of Thouless, Anderson and Palmer for p--spin interaction spin glass models is calculated. Below a critical temperature $T_c$ this number becomes exponentially large, as it is in the SK--model ($p=2$). But in contrast to this, for any $p>2$ the factor $\alpha(T)=N^{-1} \ln\langle N_s\rangle$ jumps discontinuously at $T_c(p)$, which is consistent with the discontinuity occuring within the mean--field theory for these models. For zero temperature the results obtained by Gross and M\'ezard are reproduced, and for $p\rightarrow\infty$ one gets the result for the random energy model.