A Matching Vector ($\mathbf{MV}$) family modulo a positive integer $m \ge 2$ is a pair of ordered lists $\mathcal{U} = (\mathbf{u}_1, \cdots, \mathbf{u}_K)$ and $\mathcal{V} = (\mathbf{v}_1, \cdots, \mathbf{v}_K)$ where $\mathbf{u}_i, \mathbf{v}_j \in \mathbb{Z}_m^n$ with the following property: for any $i \in [K]$, the inner product $\langle \mathbf{u}_i, \mathbf{v}_i \rangle ... more >>>

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 >>>