# Any arithmetic circuit of size $s$ and depth $\Delta$ can be converted to a formula of size $s' \leq s^{\Delta}$

I was reading Ramprasad Saptharishi's survey on Arithmetic Circuits. There in section 2.1.1 fact 2.3 it has

Any arithmetic circuit of of depth $$\Delta$$ and size $$s$$, can be simulated by an arithmetic formula of depth $$\Delta$$ and size $$s'\leq s^{\Delta}$$

I tried to do this. But i am getting $$s^s$$ how $$s^{\Delta}$$ because from node it can go to at most $$s-1$$ nodes or gates. So $$s^s$$.

It is written obviously. But i am failing to find to so. How this result is coming.

• Start from the inputs, induct on depth. If a gate has fan-out more than one, duplicate that gate a number of times equal to its fanout. It's fan-out is at most $s$. So each time you duplicate a layer of gates, you multiply the size by at most $s$. This happens at most depth many times. Nov 1, 2022 at 17:05
• (When you duplicate a gate, you duplicate the entire sub-circuit that feeds into that gate as well.) Nov 1, 2022 at 17:42