ECCC-Report TR12-121https://eccc.weizmann.ac.il/report/2012/121Comments and Revisions published for TR12-121en-usSat, 16 Mar 2013 10:46:30 +0200
Revision 2
| A note on the real $\tau$-conjecture and the distribution of roots |
Pavel Hrubes
https://eccc.weizmann.ac.il/report/2012/121#revision2
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.
Sat, 16 Mar 2013 10:46:30 +0200https://eccc.weizmann.ac.il/report/2012/121#revision2
Revision 1
| A note on the real $\tau$-conjecture and the distribution of roots |
Pavel Hrubes
https://eccc.weizmann.ac.il/report/2012/121#revision1Koiran'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.
Tue, 04 Dec 2012 20:06:59 +0200https://eccc.weizmann.ac.il/report/2012/121#revision1
Paper TR12-121
| A note on the real $\tau$-conjecture and the distribution of roots |
Pavel Hrubes
https://eccc.weizmann.ac.il/report/2012/121Koiran'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.
Tue, 25 Sep 2012 02:24:41 +0200https://eccc.weizmann.ac.il/report/2012/121