The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite the success of the theory at showing intractability of problems such as computing Brouwer fixed points and Nash ... more >>>

In this paper, we give the first subexponential (and in fact quasi-polynomial time) reconstruction algorithm for depth 3 circuits of top fan-in 3 ($\Sigma\Pi\Sigma(3)$ circuits) over the fields $\mathbb{R}$ and $\mathbb C$. Concretely, we show that given blackbox access to an $n$-variate polynomial $f$ computed by a $\Sigma\Pi\Sigma(3)$ circuit of ... more >>>

We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $\underline{\mathrm{WR}}({f}) = r$, then its Waring rank is bounded by
\[
{\mathrm{WR}}({f}) \leq d \cdot r^{O(\sqrt{r})}.
\]
This result significantly improves upon the recent bound ${\mathrm{WR}}({f}) \leq d \cdot 4^r$ established in [Dutta, Gesmundo, Ikenmeyer, Jindal, ... more >>>