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International Computer Science Symposium in Russia

CSR 2016: Computer Science – Theory and Applications pp 309–323Cite as

Depth-4 Identity Testing and Noether’s Normalization Lemma

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9691))

Abstract

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].

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Notes

  1. 1.

    We work in the projective spaces \(\mathbb {P}^n\), since by standard reason the polynomials \(Q_{i,j}\)s can be assumed to be homogeneous polynomials over \(n+1\) variables.

  2. 2.

    Notice that a zero dimensional variety contains a finite number of points.

  3. 3.

    We do not explicitly use the fact that \(Q_{i,j}\)s are irreducible in the analysis. This fact is useful if we would like to formulate a conjecture in the similar spirit of Conjecture 1 in [Gup14].

  4. 4.

    Notice that \({{\mathrm{{\small \mathrm codim}}}}(\varLambda _S) = n-(2k-1)\le n-k-1\) for \(k\ge 2\).

  5. 5.

    In the next section, we show how to construct such a subspace deterministically.

  6. 6.

    Recall that \(\deg (Q_S) \le r^{k+1}\) for any S.

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Acknowledgement

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

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Authors and Affiliations

  1. Chennai Mathematical Institute, Chennai, India

    Partha Mukhopadhyay

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  1. Partha Mukhopadhyay
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Correspondence to Partha Mukhopadhyay .

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  1. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

  2. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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Mukhopadhyay, P. (2016). Depth-4 Identity Testing and Noether’s Normalization Lemma. In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_22

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  • DOI: https://doi.org/10.1007/978-3-319-34171-2_22

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