Theory of Computing ------------------- Title : On the Real $\tau$-Conjecture and the Distribution of Complex Roots Authors : Pavel Hrubes Volume : 9 Number : 10 Pages : 403-411 URL : https://theoryofcomputing.org/articles/v009a010 Abstract -------- Koiran's real $\tau$-conjecture asserts that if a non-zero real polynomial can be written as $f=\sum_{i=1}^{p}\prod_{j=1}^{q}f_{ij}$, where each $f_{ij}$ contains at most $k$ monomials, then the number of distinct real roots of $f$ is polynomially bounded in $pqk$. We show that the conjecture implies quite a strong property of the complex roots of $f$: their arguments are uniformly distributed except for an error which is polynomially bounded in $pqk$. That is, if the conjecture is true, $f$ has degree $n$ and $f(0)\not=0$, then for every $0<\alpha-\beta< 2\pi$ |N_{\alpha,\beta}(f)- \frac{(\alpha-\beta)}{2\pi} n| \leq (pqk)^c where $c$ is an absolute constant and $N_{\alpha,\beta}(f)$ is the number of roots of $f$ of the form $r e^{i\phi }$, with $r>0$ and $\beta<\phi <\alpha$, counted with multiplicities. In particular, if the real $\tau$-conjecture is true, it is also true when multiplicities of non-zero real roots are included. A preliminary version of this paper appeared in ECCC, 2012/121.