The polarization lemma for statistical distance ($\mathrm{SD}$), due to Sahai and Vadhan (JACM, 2003), is an efficient transformation taking as input a pair of circuits $(C_0,C_1)$ and an integer $k$ and outputting a new pair of circuits $(D_0,D_1)$ such that if $\mathrm{SD}(C_0,C_1)\geq\alpha$ then $\mathrm{SD}(D_0,D_1) \geq 1-2^{-k}$ and if $\mathrm{SD}(C_0,C_1) \leq ... more >>>

In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen [Kal86, Kal87, Kal89] which shows that if an n variate degree $d$ polynomial f can be computed by an arithmetic circuit of size s, then each of its factors can ... more >>>

We say that a first-order formula $A(x_1,\dots,x_n)$ over $\mathbb{R}$ defines a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, if for every $x_1,\dots,x_n\in\{0,1\}$, $A(x_1,\dots,x_n)$ is true iff $f(x_1,\dots,x_n)=1$. We show that:

(i) every $f$ can be defined by a formula of size $O(n)$,
(ii) if $A$ is required to have at most $k\geq 1$ ... more >>>