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bio website math.brown.edu/~jhs
location Brown University Mathematics Department
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Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

14h
revised A particular interesting elliptic curve
added number theory tag
14h
answered A particular interesting elliptic curve
1d
revised Deceptive linear algebra problem
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1d
answered Deceptive linear algebra problem
Jul
30
comment What is the motivation and purpose of the Floretion group?
Actually, Carlo already gave that link in his answer on June 30.
Jul
29
answered Divisibility among discriminants
Jul
27
comment Generalized Characteristic Polynomial with Unimodular Roots
Interesting question, but could you indicate: (1) Where did these $p(z)$ polynomials arise? (2) What is the significance of unimodular condition in terms of your original problem?
Jul
19
comment Natural topologies for the space of rational functions
Okay, but since it's dealing with iteration, and the degree of $f^n$ is $(\deg f)^n$, somehow it needs to be relating maps of differing degrees. In any case, Laura would be a good person to ask about this.
Jul
19
comment Growth of the size of iterated polynomials
The section in my book says that the limit $$\lim d^{-n}\log^+|p^n(a)|$$ converges, and differs from $\log^+|a|$ by a quantity that is bounded solely in terms of $p$. If you exponentiate that relation, you get the statement in my comment.
Jul
19
comment Natural topologies for the space of rational functions
Not sure if it's quite what you need, but there's a paper by Laura DeMarco, "Iteration at the boundary of the space of rational maps", Duke Math. Journal. 130 (2005) 169-197, that might be relevant. In any case, I seem to recall that she considers some sort of inductive (projective?) limit of the space of rational functions of degree $d$ over all $d$. The paper is available here: math.northwestern.edu/~demarco/Duke_boundary.pdf
Jul
19
revised Natural topologies for the space of rational functions
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Jul
19
comment Natural topologies for the space of rational functions
@GeraldEdgar It wasn't Arnold that made that remark. (Not that it matters.)
Jul
19
comment Natural topologies for the space of rational functions
As long as you avoid letting $a\to0$, you can just take the distance from $a/(b-z)$ to $a'/(b'-z)$ to be $|b/a-b'/a'|$, I think. But if you want a nice metric topology that's okay for all $(a,b)\in\mathbb C^2$, there may well be a serious problem in the neighborhood the line $a=0$.
Jul
19
answered Natural topologies for the space of rational functions
Jul
18
comment Equidistribution of representations by a binary cubic form
@GHfromMO Yes, of course you're right, I left off the log. Unfortunately, I can't edit the comment, but you're comment clears it up. Thanks.
Jul
18
comment Equidistribution of representations by a binary cubic form
@GHfromMO You're welcome. This also reminds me of a related question that I believe is still open, namely whether there is even a single $f$ such that there is an infinite sequence of $m$'s for which the number of solutions to $f(X,Y)=m$ is ${}\gg|m|$. (And since some people have been suggesting that there is an absolute upper bound for the rank of elliptic curves over $\mathbb Q$, it's not even clear that the elementary method in my paper will get you to ${}\gg|m|^{1-\epsilon}$ for every $\epsilon>0$.)
Jul
17
answered Equidistribution of representations by a binary cubic form
Jul
17
answered Intuition behind Kronecker's congruence?
Jul
16
comment Growth of the size of iterated polynomials
@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
comment Growth of the size of iterated polynomials
@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.