  Under the auspices of the Computational Complexity Foundation (CCF)     REPORTS > KEYWORD > DEPTH 3 ARITHMETIC CIRCUITS:
Reports tagged with Depth 3 arithmetic circuits:
TR07-095 | 13th July 2007

#### The Ideal Membership Problem and Polynomial Identity Testing

Revisions: 2

\begin{abstract}
Given a monomial ideal $I=\angle{m_1,m_2,\cdots,m_k}$ where $m_i$
are monomials and a polynomial $f$ as an arithmetic circuit the
\emph{Ideal Membership Problem } is to test if $f\in I$. We study
this problem and show the following results.
\begin{itemize}
\item[(a)] If the ideal $I=\angle{m_1,m_2,\cdots,m_k}$ for a
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TR18-068 | 8th April 2018
Mrinal Kumar

#### On top fan-in vs formal degree for depth-3 arithmetic circuits

Revisions: 1

We show that over the field of complex numbers, every homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, ... more >>>

TR18-111 | 4th June 2018
Vikraman Arvind, Abhranil Chatterjee, Rajit Datta, Partha Mukhopadhyay

#### Beating Brute Force for Polynomial Identity Testing of General Depth-3 Circuits

Let $C$ be a depth-3 $\Sigma\Pi\Sigma$ arithmetic circuit of size $s$,
computing a polynomial $f \in \mathbb{F}[x_1,\ldots, x_n]$ (where $\mathbb{F}$ = $\mathbb{Q}$ or
$\mathbb{C}$) with fan-in of product gates bounded by $d$. We give a
deterministic time $2^d \text{poly}(n,s)$ polynomial identity testing
algorithm to check whether $f \equiv 0$ or ... more >>>

TR21-155 | 13th November 2021
Vishwas Bhargava, Ankit Garg, Neeraj Kayal, Chandan Saha

#### Learning generalized depth-three arithmetic circuits in the non-degenerate case

Consider a homogeneous degree $d$ polynomial $f = T_1 + \cdots + T_s$, $T_i = g_i(\ell_{i,1}, \ldots, \ell_{i, m})$ where $g_i$'s are homogeneous $m$-variate degree $d$ polynomials and $\ell_{i,j}$'s are linear polynomials in $n$ variables. We design a (randomized) learning algorithm that given black-box access to $f$, computes black-boxes for ... more >>>

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