Volume 9 (2013) Article 10 pp. 403-411
On the Real $\tau$-Conjecture and the Distribution of Complex Roots
Revised: April 11, 2013
Published: May 21, 2013
$\newcommand{\eee}{\mathrm{e}}$
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 $0<\alpha-\beta< 2\pi$ $\Big|N_{\alpha,\beta}(f)- \frac{(\alpha-\beta)}{2\pi} n \Big|\leq (pqk)^c\,,$where $c$ is an absolute constant and $N_{\alpha,\beta}(f)$ is the number of roots of $f$ of the form $r\eee^{i\phi }$, 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.