Rectangular matrix multiplication from polynomial products

In this note, we extend the matrix-multiplication analysis by OpenAI to rectangular products and prove $ω(1,k,1)\le2$ for $0\le k\le\frac{1}{2}$, and $ω(1,k,1)\le1+k+\frac{1}{4k}$ for $k\ge\frac{1}{2}$. In particular, $ω(1, \frac{1}{2},1)=2$ and the dual exponent satisfies $α\ge\frac{1}{2}$. The pro