Timeline for Small values of a polynomial evaluated at roots of unity
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2014 at 0:26 | comment | added | Douglas Lind | Yes, real dimension as a subset of the $d$-torus, which has real dimension $d$. And yes, this covers "most" polynomials. The concrete case above is where all our techniques fail. | |
| Aug 21, 2014 at 15:54 | comment | added | Joe Silverman | Thanks for the reference. I was looking at the earlier paper, where you'd assumed that the intersection is finite (which already looks quite intricate). I'll take a look at the 1108.4989 paper. You're talking about real dimension, right? So $f=0$ on $(\mathbb{C}^*)^d$ has real codimension 2, so generically it should intersect $(S^1)^d$ in a set of real codimension 2. This would seem to indicate that your result applies to "most polynomials" in some appropriate sense. Or am I confused about codimension here? | |
| Aug 21, 2014 at 15:39 | comment | added | Douglas Lind | In arxiv.org/abs/1108.4989 we proved that the averages converge provided that the zero set of $f$ on the $n$-torus has dimension at most $n-2$ (this is a successor to the paper @Andreas mentions). The argument uses homoclinic points for the associated algebraic action, which we can create with the assumption on the zero set by finding another polynomial, not a multiple of $f$, whose zero set on the torus contains the zero set of $f$ on the torus. This is "atonality" as introduced by Agler, McCarthy, and Stankus in their work on function theory on polydisks. | |
| Aug 21, 2014 at 15:10 | comment | added | Joe Silverman | Thanks, Doug. Actually, the "related, but different" problem is the one I'm interested in! But I thought that the question that I posed was likely to be easier, and that it would be an essential step in proving the convergence to log Mahler measure. That's interesting that you say the question is motivated by a dynamical question about periodic points. Now that you say that, it seems clear, but I came at it from the viewpoint of torsion points on algebraic groups. Do you know any other references or discussion of the problem of $\hbox{Avg}\log_0|f|\to\log M(f)$? | |
| Aug 21, 2014 at 14:57 | history | answered | Douglas Lind | CC BY-SA 3.0 |