# 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},
journal={ArXiv},
year={2014},
volume={abs/1401.0189}
}
• Published 31 December 2013
• Mathematics, Computer Science
• ArXiv
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, Mathematics
SIAM 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, Mathematics
2014 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 Science
Electron. 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
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.

## References

SHOWING 1-10 OF 27 REFERENCES
Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach
• Computer Science, Mathematics
STACS
• 2014
A simple combinatorial property is identified such that any polynomial f that satisfies the property would achieve similar circuit size lower bound for depth-4 circuits, and it does not matter whether f is in VP or in VNP.
On the Power of Homogeneous Depth 4 Arithmetic Circuits
• Computer Science, Mathematics
2014 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.
Lower bounds for depth 4 formulas computing iterated matrix multiplication
• Mathematics, Computer Science
STOC
• 2014
It is shown that any multilinear homogeneous depth 4 arithmetic formula computing the product of d generic matrices of size n × n, IMMn,d, has size nΩ(√d) as long as d = nO(1), improving the result of Nisan and Wigderson (Computational Complexity, 1997).
Arithmetic Circuit Lower Bounds via MaxRank
• Computer Science, Mathematics
ICALP
• 2013
The polynomial coefficient matrix is introduced and maximum rank of this matrix under variable substitution is identified as a complexity measure for multivariate polynomials and super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits are proved.
An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas
• Computer Science
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
• 2014
This work gives an explicit family of polynomials of degree d on N variables with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Σi,j, it must hold that 2Ω(√d·log N) ≥ 2.
Arithmetic Circuits: A Chasm at Depth Three
• Computer Science, Mathematics
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
We show that, over Q, if an n-variate polynomial of degree d = n<sup>O(1)</sup> is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it
Arithmetic circuits: The chasm at depth four gets wider
• P. Koiran
• Mathematics, Computer Science
Theor. Comput. Sci.
• 2012
A super-polynomial lower bound for regular arithmetic formulas
• Mathematics, Computer Science
STOC
• 2013
It is shown that there is an n2-variate polynomial of degree n in VNP such that any regular formula computing it must be of size at least nΩ(log n) and any further asymptotic improvement in the lower bound for such formulas will imply that VP is different from VNP.
Diagonal Circuit Identity Testing and Lower Bounds
In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonaldepth-3 circuit C(x 1 ,...,x n ) (i.e. Cis a sum of powers of linear functions) is zero. We also