On the Real τ-Conjecture and the Distribution of Complex Roots

Koiran’s real τ-conjecture asserts that if a non-zero real polynomial can be written as f = ∑ i=1 ∏ q j=1 fi j, where each fi j 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) 6= 0, then for every 0 < α−β < 2π ∣∣∣Nα,β ( f )− (α−β ) 2π n∣∣∣≤ (pqk)c , where c is an absolute constant and Nα,β ( f ) is the number of roots of f of the form reiφ , with r > 0 and β < φ < α , counted with multiplicities. In particular, if the real τ-conjecture is true, it is also true when multiplicities of non-zero real roots are included. ACM Classification: F.1.m AMS Classification: 03D15, 68Q17