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Electronic Colloquium on Computational Complexity

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Reports tagged with Arithmetic Circuits:
TR22-022 | 18th February 2022
Vikraman Arvind, Pushkar Joglekar

On Efficient Noncommutative Polynomial Factorization via Higman Linearization

Revisions: 3

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F of polynomials in noncommuting variables x_1,x_2,…,x_n over the field F. We obtain the following result:

Given a noncommutative arithmetic formula of size s computing a noncommutative polynomial f in F as input, where ... more >>>

TR22-042 | 31st March 2022
Vikraman Arvind, Pushkar Joglekar

Matrix Polynomial Factorization via Higman Linearization

In continuation to our recent work on noncommutative
polynomial factorization, we consider the factorization problem for
matrices of polynomials and show the following results.
\item Given as input a full rank $d\times d$ matrix $M$ whose entries
$M_{ij}$ are polynomials in the free noncommutative ring
more >>>

TR22-151 | 12th November 2022
Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, Bhargav Thankey

Low-depth arithmetic circuit lower bounds via shifted partials

We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving ... more >>>

TR24-080 | 16th April 2024
Robert Andrews, Avi Wigderson

Constant-Depth Arithmetic Circuits for Linear Algebra Problems

We design polynomial size, constant depth (namely, $AC^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, Bézout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and Bézout matrices. Our GCD algorithm extends to any ... more >>>

ISSN 1433-8092 | Imprint