All reports by Author Pushkar Joglekar:

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TR23-025
| 10th March 2023
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Vikraman Arvind, Pushkar Joglekar#### Multivariate to Bivariate Reduction for Noncommutative Polynomial Factorization

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TR22-042
| 31st March 2022
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Vikraman Arvind, Pushkar Joglekar#### Matrix Polynomial Factorization via Higman Linearization

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TR22-022
| 18th February 2022
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Vikraman Arvind, Pushkar Joglekar#### On Efficient Noncommutative Polynomial Factorization via Higman Linearization

Revisions: 3

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TR17-087
| 9th May 2017
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Pushkar Joglekar, Raghavendra Rao B V, Sidhartha Sivakumar#### On Weak-Space Complexity over Complex Numbers

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TR16-193
| 22nd November 2016
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Vikraman Arvind, Pushkar Joglekar, Partha Mukhopadhyay, Raja S#### Identity Testing for +-Regular Noncommutative Arithmetic Circuits

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TR15-141
| 26th August 2015
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Pushkar Joglekar, Aravind N.R.#### On the expressive power of read-once determinants

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TR15-124
| 3rd August 2015
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Vikraman Arvind, Pushkar Joglekar, Raja S#### Noncommutative Valiant's Classes: Structure and Complete Problems

Revisions: 1

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TR15-004
| 4th January 2015
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Vikraman Arvind, Pushkar Joglekar, Gaurav Rattan#### On the Complexity of Noncommutative Polynomial Factorization

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TR09-073
| 6th September 2009
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Vikraman Arvind, Pushkar Joglekar, Srikanth Srinivasan#### On Lower Bounds for Constant Width Arithmetic Circuits

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TR09-026
| 17th February 2009
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Vikraman Arvind, Pushkar Joglekar#### Arithmetic Circuit Size, Identity Testing, and Finite Automata

Vikraman Arvind, Pushkar Joglekar

Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following:

(1) In the white-box setting, given an n-variate noncommutative polynomial f in F over a field F (either a ... more >>>

Vikraman Arvind, Pushkar Joglekar

In continuation to our recent work on noncommutative

polynomial factorization, we consider the factorization problem for

matrices of polynomials and show the following results.

\begin{itemize}

\item Given as input a full rank $d\times d$ matrix $M$ whose entries

$M_{ij}$ are polynomials in the free noncommutative ring

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Vikraman Arvind, Pushkar Joglekar

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 >>>

Pushkar Joglekar, Raghavendra Rao B V, Sidhartha Sivakumar

Defining a feasible notion of space over the Blum-Shub-Smale (BSS) model of algebraic computation is a long standing open problem. In an attempt to define a right notion of space complexity for the BSS model, Naurois [CiE, 2007] introduced the notion of weak-space. We investigate the weak-space bounded computations and ... more >>>

Vikraman Arvind, Pushkar Joglekar, Partha Mukhopadhyay, Raja S

An efficient randomized polynomial identity test for noncommutative

polynomials given by noncommutative arithmetic circuits remains an

open problem. The main bottleneck to applying known techniques is that

a noncommutative circuit of size $s$ can compute a polynomial of

degree exponential in $s$ with a double-exponential number of nonzero

monomials. ...
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Pushkar Joglekar, Aravind N.R.

We introduce and study the notion of read-$k$ projections of the determinant: a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$ is called a {\it read-$k$ projection of determinant} if $f=det(M)$, where entries of matrix $M$ are either field elements or variables such that each variable appears at most $k$ times in ... more >>>

Vikraman Arvind, Pushkar Joglekar, Raja S

In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and

$\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some

striking connections to classical formal language theory. Our main

results are the following:

(1) We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for ... more >>>

Vikraman Arvind, Pushkar Joglekar, Gaurav Rattan

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\mathbb{F}\langle x_1,x_2,\ldots,x_n\rangle$ of polynomials over the field $\mathbb{F}$ and noncommuting variables $x_1,x_2,\ldots,x_n$. Our main results are the following.

Although $\mathbb{F}\langle x_1,\dots,x_n \rangle$ is not a unique factorization ring, we note that variable-disjoint factorization in ... more >>>

Vikraman Arvind, Pushkar Joglekar, Srikanth Srinivasan

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following.

1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit ...
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Vikraman Arvind, Pushkar Joglekar

Let $\F\{x_1,x_2,\cdots,x_n\}$ be the noncommutative polynomial

ring over a field $\F$, where the $x_i$'s are free noncommuting

formal variables. Given a finite automaton $\A$ with the $x_i$'s as

alphabet, we can define polynomials $\f( mod A)$ and $\f(div A)$

obtained by natural operations that we ...
more >>>