LetX1, . . . , Xn be indeterminates over Q and letX := (X1, . . . , Xn). Let F1, . . . , Fp be a regular sequence of polynomials in Q[X] of degree at most d such that for each 1 ≤ k ≤ p the ideal (F1, . . . , Fk) is radical. Suppose that the variables X1, . . . , Xn are in generic position with respect to F1, . . . , Fp. Further suppose that the polynomials are given by an essentially division-free circuit β in Q[X] of size L and non-scalar depth `. We present a family of algorithms Πi and invariants δi of F1, . . . , Fp, 1 ≤ i ≤ n− p, such that Πi produces on input β a smooth algebraic sample point for each connected component of {x ∈ Rn | F1(x) = · · · = Fp(x) = 0} where the Jacobian of F1 = 0, . . . , Fp = 0 has generically rank p. The sequential complexity of Πi is of order L(nd)O(1)(min{(nd)c n, δi})2 and its non-scalar parallel complexity is of order O(n(`+ lognd) log δi). Here c > 0 is a suitable universal constant. Thus, the complexity of Πi meets the already known worst case bounds. The particular feature of Πi is its pseudo-polynomial and intrinsic complexity character and this entails the best runtime behavior one can hope for. The algorithm Πi works in the non-uniform deterministic as well as in the uniform probabilistic complexity model. We exhibit also a worst case estimate of order (nn d)O(n) for the invariant δi. The reader may notice that this bound overestimates the extrinsic complexity of Πi, which is bounded by (nd)O(n).