Timeline for Growth of the size of iterated polynomials
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 18, 2015 at 16:38 | vote | accept | Spock | ||
| Jul 16, 2015 at 21:01 | comment | added | Joe Silverman | @LasseRempe-Gillen Yes, the "logarithmic height of 0 is 0, not $-\infty$". See comment answering Alexandre, $h(0)=h(0/1)$. Anyway, yes, the idea of using a telescoping sum undoubtedly predates Brolin and Tate. I guess I opened a can of worms by giving Tate credit. I withdraw that assignment for polynomials of one variable. But Tate realized that one can combine this telescoping sum trick with the height machinery that Weil had developed and divisor class relations in algebraic geometry to create height functions that transform nicely in quite general contexts, especially on abelian varieties. | |
| Jul 16, 2015 at 20:57 | comment | added | Joe Silverman | @AlexandreEremenko No, it's not. But the elliptic curve case is precisely iteration of a Lattes map, so admittedly not a polynomial, but it is a rational function. Of course, Lattes maps are quite special, but I have a recollection of seeing somewhere (in a paper of Lang's maybe) a comment that Tate had told him that one could do the same thing with $d^{-n}h(f^n(x))$, where $f$ is a rational function of degree $d\ge2$ and $h$ denotes the logarithmic height, so for a rational number $a/b$, it's $h(a/b)=\log\max\{|a|,|b|\}$. But it's a different context from what Brolin was doing. | |
| Jul 16, 2015 at 20:55 | comment | added | Lasse Rempe | I am a little confused - but may have missed some subtlety. 1) Should $\log$ not be $\log_+$ (i.e., maximum of 0 and the logarithm)? Otherwise, what happens for $p(z)=z^2$ and $a=0$? 2) If this is the case, why isn't the convergence a consequence of [the proof of] Boettcher's theorem? | |
| Jul 16, 2015 at 18:43 | comment | added | Alexandre Eremenko | Is dynamics (iteration of a polynomial) really mentioned in Manin's paper you refer to? | |
| Jul 16, 2015 at 11:57 | comment | added | Joe Silverman | ... it would be fair to say that Tate worked out the theory of such limits in the context of number theory/algebraic geometry, while Brolin studied and used such limits in the context of (complex) dynamical systems. And actually, the Brolin functions, which would be defined for all complex points, are what in number theory are called "local height functions", in his case for an archimedean absolute value. There are also $p$-adic local heights, and their sum gives the global height. The construction of local heights for abelian varieties is due to Neron, also circa 1964/1965. | |
| Jul 16, 2015 at 11:54 | comment | added | Joe Silverman | @AlexandreEremenko Tate constructed canonical heights on elliptic curves by using iteration of the doubling map on the $x$-coordinate (so Lattes maps, although I don't think he knew about Lattes work). But actually, his construction is much more general, on abelian varieties. However, comme d'habitude, Tate never actually published this material. It was first described, I believe, by Manin (The Tate height of points on an abelian variety..., Izv Akad Nauk SSR Ser Math 28 (1964)). When Tate worked it out, I don't know. But I don't want to get into a priority argument, so maybe ... | |
| Jul 16, 2015 at 7:23 | comment | added | Alexandre Eremenko | "The fact that this converges is due to Tate", can you give the year, or a reference? The fact that this converges in the complex domain is due to Hans Brolin (1965) :-) | |
| Jul 15, 2015 at 22:34 | history | answered | Joe Silverman | CC BY-SA 3.0 |