# On top fan-in vs formal degree for depth-3 arithmetic circuits

@article{Kumar2018OnTF, title={On top fan-in vs formal degree for depth-3 arithmetic circuits}, author={Mrinal Kumar}, journal={Electron. Colloquium Comput. Complex.}, year={2018}, volume={TR18} }

We show that over the field of complex numbers, \emph{every} homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, such that any depth-$3$ arithmetic circuit computing $P$ must have top fan-in at least $\Omega(n)$.
As an application, we get a new tradeoff between the top fan-in and formal degree…

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