Depth-4 Identity Testing and Noether’s Normalization Lemma

  • Partha MukhopadhyayEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


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 \(k,r=O(1)\). 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].



I thank K.V. Subrahmanyam for many helpful discussions.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Chennai Mathematical InstituteChennaiIndia

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