Abstract. Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1, . . . , fm} ⊂ F[x1, . . . , xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper we design blackbox and efficient linear maps ϕ that reduce the number of variables from n to r but maintain trdeg{ϕ(fi)}i = r, assuming fi's sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing:1. Given a circuit C and sparse subcircuits f1, . (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.

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