Let $p(x_1,...,x_n) =\sum_{ (r_1,...,r_n) \in I_{n,n} } a_{(r_1,...,r_n) } \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$
be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients.
The support of such polynomial $p(x_1,...,x_n)$
is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 ... more >>>

We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators $G : \F^s \to \F^n$ that fool polynomials over a prime field $\F$:
\begin{enumerate}
\item a ... more >>>

We prove that the sum of $d$ small-bias generators $L
: \F^s \to \F^n$ fools degree-$d$ polynomials in $n$
variables over a prime field $\F$, for any fixed
degree $d$ and field $\F$, including $\F = \F_2 =
{0,1}$.

Our result improves on both the work by Bogdanov and
Viola ... more >>>

We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W[P]-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Computing, 38(4), 2008).

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We report on some initial results of a brute-force search for determining the maximum correlation between degree-$d$ polynomials modulo $p$ and the $n$-bit mod $q$ function. For various settings of the parameters $n,d,p,$ and $q$, our results indicate that symmetric polynomials yield the maximum correlation. This contrasts with the previously-analyzed ... more >>>

We highlight the special case of Valiant's rigidity
problem in which the low-rank matrices are truth-tables
of sparse polynomials. We show that progress on this
special case entails that Inner Product is not computable
by small $\acz$ circuits with one layer of parity gates
close to the inputs. We then ... more >>>

We show that degree-$d$ block-symmetric polynomials in
$n$ variables modulo any odd $p$ correlate with parity
exponentially better than degree-$d$ symmetric
polynomials, if $n \ge c d^2 \log d$ and $d \in [0.995
\cdot p^t - 1,p^t)$ for some $t \ge 1$. For these
infinitely many degrees, our result ... more >>>

We draw two incomplete, biased maps of challenges in
computational complexity lower bounds. Our aim is to put
these challenges in perspective, and to present some
connections which do not seem widely known.

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We study the complexity of detecting monomials
with special properties in the sum-product
expansion of a polynomial represented by an arithmetic
circuit of size polynomial in the number of input
variables and using only multiplication and addition.
We focus on monomial properties expressed in terms
of the number of distinct ... more >>>

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorthmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with
boolean matrices with prescribed row and (uniformly bounded) column sums within simply ... more >>>

We show that for any coprime $m,r$ there is a circuit of the form $\text{MOD}_m\circ \text{AND}_{d(n)}$ whose correlation with $\text{MOD}_r$ is at least $2^{-O\left( \frac{n}{d(n)} \right) }$. This is the first correlation lower bound for arbitrary $m,r$, whereas previously lower bounds were known for prime $m$. Our motivation is the ... more >>>

We construct pseudorandom generators with improved seed length for
several classes of tests. First we consider the class of read-once
polynomials over GF(2) in $m$ variables. For error $\e$ we obtain seed
length $\tilde O (\log(m/\e)) \log(1/\e)$, where $\tilde O$ hides lower-order
terms. This is optimal up to the factor ... more >>>

We give the first polynomial-time *non-adaptive* proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. Our algorithm, for $s$-sparse polynomial over $n$ variables, makes $q=(s/\epsilon)^{\gamma(s,\epsilon)}\log n$ queries where $2.66\le \gamma(s,\epsilon)\le 6.922$ and runs in $\tilde O(n)\cdot poly(s,1/\epsilon)$ time. We also show that for any $\epsilon=1/s^{O(1)}$ any non-adaptive ... more >>>