Vikraman Arvind, Partha Mukhopadhyay

\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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Mrinal Kumar

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 >>>

Vikraman Arvind, Abhranil Chatterjee, Rajit Datta, Partha Mukhopadhyay

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 ...
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