# Arithmetic Circuit Lower Bounds via MaxRank

@article{Kumar2013ArithmeticCL, title={Arithmetic Circuit Lower Bounds via MaxRank}, author={Mrinal Kumar and Gaurav Maheshwari and Jayalal Sarma}, journal={ArXiv}, year={2013}, volume={abs/1302.3308} }

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results :
· As our first main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension n ×n requires Ω(nd−1/2d…

## 7 Citations

Lower Bounds for Projections of Power Symmetric Polynomials

- Mathematics, Computer Science
- 2016

Lower bounds imply deterministic npoly(logn) black-box identity testing algorithms for the above classes of arithmetic circuits.

On the limits of depth reduction at depth 3 over small finite fields

- Mathematics, Computer ScienceInf. Comput.
- 2017

Limitations of sum of products of Read-Once Polynomials

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

A class of formulas of unbounded depth with exponential size lower bound against the permanent can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz for a sub-class of multi-linear and non-multi-linear formulas.

On Constant Depth Circuits Parameterized by Degree: Identity Testing and Depth Reduction

- MathematicsCOCOON
- 2017

The notion of fixed parameter tractability is defined and it is shown that there are families of polynomials of degree k that cannot be computed by homogeneous depth four \(\varSigma \varPi ^{\sqrt{k}}\varS Sigma \var Pi ^{k}\) circuits, implying that there is no parameterized depth reduction for circuits of size f(k)n^{O(1) such that the resulting depth four circuit is homogeneous.

A note on parameterized polynomial identity testing using hitting set generators

- MathematicsInf. Process. Lett.
- 2019

Towards an algebraic natural proofs barrier via polynomial identity testing

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2017

It is observed that a certain kind of algebraic proof cannot be used to prove lower bounds against VP if and only if what the authors call succinct hitting sets exist for VP.

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