Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials \{f_1,\ldots, f_m\} \subset \mathbb{F}[x_1,\ldots, x_n] are called algebraically independent if there is no non-zero polynomial F such that F(f_1, \ldots, f_m) = 0. The transcendence degree, \mbox{trdeg}\{f_1,\ldots, f_m\}, is the maximal number r of algebraically independent polynomials in the set. In this paper we design blackbox and efficient linear maps \varphi that reduce the number of variables from n to r but maintain \mbox{trdeg}\{\varphi(f_i)\}_i = r, assuming f_i's sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing:

(1) Given a polynomial-degree circuit C and sparse polynomials f_1,\ldots, f_m with trdeg r, we can test blackbox D := C(f_1, \ldots, f_m) for zeroness in \mbox{poly}(\mbox{size}(D))^r time.
(2) Define a \Sigma\Pi\Sigma\Pi_\delta(k,s,n) circuit C to be of the form \sum_{i=1}^k \prod_{j=1}^s f_{i,j}, where f_{i,j} are sparse n-variate polynomials of degree at most \delta. For k = 2 we give a \mbox{poly}(\delta sn)^{\delta^2} time blackbox identity test.
(3) For a general depth-4 circuit we define a notion of rank. Assuming there is a rank bound R for minimal simple \Sigma\Pi\Sigma\Pi_\delta(k,s,n) identities, we give a \mbox{poly}(\delta snR)^{Rk\delta^2} time blackbox identity test for \Sigma\Pi\Sigma\Pi_\delta(k,s,n) circuits. This partially generalizes the state of the art of depth-3 to depth-4 circuits.

The notion of trdeg works best with large or zero characteristic, but we also give versions of our results for arbitrary fields.