Given query access to a monotone function $f\colon\{0,1\}^n\to\{0,1\}$ with certificate complexity $C(f)$ and an input $x^{\star}$, we design an algorithm that outputs a size-$C(f)$ subset of $x^{\star}$ certifying the value of $f(x^{\star})$. Our algorithm makes $O(C(f) \cdot \log n)$ queries to $f$, which matches the information-theoretic lower bound for this ... more >>>

We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there ... more >>>

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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