In (ToCT’20) Kumar surprisingly proved that every polynomial can be approximated as a sum of a constant and a product of linear polynomials. In this work, we prove the converse of Kumar's result which ramifies in a surprising new formulation of Waring rank and border Waring rank. From this conclusion, ... more >>>

Given polynomials $f,g,h\,\in \mathbb{F}[x_1,\ldots,x_n]$ such that $f=g/h$, where both $g$ and $h$ are computable by arithmetic circuits of size $s$, we show that $f$ can be computed by a circuit of size $\poly(s,\deg(h))$. This solves a special case of division elimination for high-degree circuits (Kaltofen'87 \& WACT'16). The result ... more >>>

Algebraic natural proofs were recently introduced by Forbes, Shpilka and Volk (Proc. of the 49th Annual {ACM} {SIGACT} Symposium on Theory of Computing (STOC), pages {653--664}, 2017) and independently by Grochow, Kumar, Saks and Saraf~(CoRR, abs/1701.01717, 2017) as an attempt to transfer Razborov and Rudich's famous barrier result (J. Comput. ... more >>>

We consider the problem of commutative rank computation of a given matrix space, $\mathcal{B}\subseteq\mathbb{F}^{n\times n}$. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is $n$, subsumes problems such as testing perfect matching in graphs ... more >>>