We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One only need to consider size- s degree- s circuits that depend on the first log c s variables (where c is a constant and we are composing c logarithms). Thus, hitting-set generator (hsg) manifests a {\em bootstrapping} behavior--- a partial hsg against very few variables can be efficiently grown to a complete hsg. A boolean analog, or a pseudorandom generator property of this type, is unheard-of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially wrt variables. This is repeated c times in an efficient way.

Pushing the envelope further we show that: {\bf (1)} a quadratic-time blackbox PIT for 6913 -variate degree- s size- s polynomials, will lead to a ``near''-complete derandomization of PIT, and {\bf (2)} a blackbox PIT for n -variate degree- s size- s circuits in s n -time, for 1 2 , will lead to a ``near''-complete derandomization of PIT (in contrast, s n -time is trivial).

Our second idea is to study depth- 4 circuits that depend on constantly many variables. We show that a polynomial-time computable, O ( s 1 49 ) -degree hsg for {\em trivariate} depth- 4 circuits bootstraps to a quasipolynomial time hsg for general poly-degree circuits, and implies a lower bound that is a bit stronger than Kabanets-Impagliazzo (STOC 2003).