### Revision(s):

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Revision #2 to TR12-121 | 16th March 2013 10:46
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#### A note on the real $\tau$-conjecture and the distribution of roots

**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 polynomial 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 polynomial in $pqk$. That is, if the conjecture is true, $f$ has degree $n$ and $f(0)\not=0$, then

for every $00$ 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.

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Revision #1 to TR12-121 | 4th December 2012 20:06
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#### A note on the real $\tau$-conjecture and the distribution of roots

**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 polynomial 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 polynomial in $pqk$. In particular, if the real $\tau$-conjecture is true it also true when multiplicities of real roots are included.

### Paper:

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TR12-121 | 25th September 2012 00:16
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#### A note on the real $\tau$-conjecture and the distribution of roots

**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 polynomial 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 polynomial in $pqk$. In particular, if the real $\tau$-conjecture is true it also true when multiplicities of real roots are included.