SIGMA-MODELS WITH A(K) SINGULARITIES IN EUCLIDEAN SPACETIME OF DIMENSION 0-LESS-THAN-OR-EQUAL-TO-D-LESS-THAN-4 AND IN THE LIMIT N-]INFINITY

For the case of the single-O(N)-vector linear sigma models the critica l behaviour following from any A(k) singularity in the action is worke d out in the double scaling limit N --> infinity, f(r) --> f(r)(c), 2 less than or equal to r less than or equal to k. After an exact elimin ation of Gaussian degrees of freedom, the critical objects such as cou pling constants, indices and susceptibility matrix are derived for all A(k) and spacetime dimensions 0 less than or equal to D < 4. There ap pear exceptional spacetime dimensions where the degree k of the singul arity A(k) is more strongly constrained than by the renormalizability requirement.