Depth-4 Identity Testing and Noether's Normalization Lemma

We consider the black-box polynomial identity testing $${\small \mathrm {PIT}}$$ problem for a sub-class of depth-4 $$\varSigma \varPi \varSigma \varPi k,r$$ circuits. Such circuits compute polynomials of the following type: $$ C{\small \mathrm X} = \sum _{i=1}^k \prod _{j=1}^{d_i} Q_{i,j}, $$ where k is the fan-in of the top $$\varSigma $$ gate and r is the maximum degree of the polynomials $$\{Q_{i,j}\}_{i\in [k], j\in [d_i]}$$ , and $$k,r=O1$$ . We consider a sub-class of such circuits satisfying a generic algebraic-geometric restriction, and we give a deterministic polynomial-time black-box $${\small \mathrm {PIT}}$$ algorithm for such circuits. Our study is motivated by two recent results of Mulmuley FOCS 2012, [Mul12], and Gupta ECCC 2014, [Gup14]. In particular, we obtain the derandomization by solving a particular instance of derandomization problem of Noether's Normalization Lemma $$\mathrm{NNL}$$ . Our result can also be considered as a unified way of viewing the depth-4 $${\small \mathrm {PIT}}$$ problems closely related to the work of Gupta [Gup14], and the approach suggested by Mulmuley [Mul12].