# On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

@article{Chillara2014OnTL, title={On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields}, author={Suryajith Chillara and Partha Mukhopadhyay}, journal={ArXiv}, year={2014}, volume={abs/1401.0189} }

In a surprising recent result, Gupta et al. [GKKS13b] have proved that over ℚ any n O(1)-variate and n-degree polynomial in VP can also be computed by a depth three ∑ ∏ ∑ circuit of size \(2^{O(\sqrt{n} \log^{3/2}n)}\) . Over fixed-size finite fields, Grigoriev and Karpinski proved that any ∑ ∏ ∑ circuit that computes the determinant (or the permanent) polynomial of a n×n matrix must be of size 2Ω(n). In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that…

## 4 Citations

On the Power of Homogeneous Depth 4 Arithmetic Circuits

- Computer Science, MathematicsSIAM J. Comput.
- 2017

In this work, exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$ are proved and the first exponential separation between general arithmetic circuits and homogeneousDepth4 arithmetic circuits is given.

On the Power of Homogeneous Depth 4 Arithmetic Circuits

- Computer Science, Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

It is shown that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension nO(1) must have size nΩ(√n) and the results strengthen previous works in two significant ways.

An exponential lower bound for homogeneous depth-5 circuits over finite fields

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2015

This paper shows exponential lower bounds for the class of homogeneous depth-$5 circuits over all small finite fields and builds over a tighter analysis of the lower bound of Kumar and Saraf [KS14], the first super-polynomial lower bound for this class for any field.

Lower bounds for bounded depth arithmetic circuits

- Computer Science, Mathematics
- 2017

A strong hierarchy theorem for bottom fan-in for homogeneous depth-4 circuits, a superpolynomial lower bound for homogeneity depth-5 circuits over finite fields, and some results indicating that the parameters for depth reduction to homogeneous Depth 4 circuits might be close to optimal in a fairly strong sense.

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