11,378 reputation
3370
bio website math.brown.edu/~jhs
location Brown University Mathematics Department
age
visits member for 3 years, 8 months
seen 1 hour ago
Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

17h
awarded  nt.number-theory
1d
answered Ideal classes fixed by the Galois group
2d
comment Separation of lattice points on the Mordell elliptic curve
@RyanD'Mello Actually, if you assume that $(x,y)$ is a solution with $x$ reasonably large and $|x^3-y^2|\asymp\sqrt{|x|}$, and if you let $(x+\ell,y+k)$ be the next largest solution, then approximating $(x+\ell)^3\approx x^3+3\ell x^2$ and $(y+k)^2\approx y^2+2ky$ gives $k$ in terms of $\ell$ (more or less), and then looking at the next largest terms may well give a lower bound for $\ell$ in terms of $x$. I'll let you work out the details (and maybe this doesn't work at all).
2d
comment Separation of lattice points on the Mordell elliptic curve
@RyanD'Mello Please don't misquote me. I didn't say that there is no simple answer using currently known (or even possibly quite elementary) methods. I simply said that at a quick glance, the results/methods in my paper don't seem to yield anything, and that offhand I don't recall seeing this problem addressed in the literature (which is a much weaker statement than saying that it hasn't been considered before!).
Aug
26
comment Separation of lattice points on the Mordell elliptic curve
@RyanD'Mello Offhand I don't know what lower bound on $x_2-x_1$ can be regarded as trivial and what would be significant. The best way to figure that out is to try to prove something, which I'll let you do, rather than my working more on the problem.
Aug
26
comment Composition of a transcendental function with a rational function
By assumption, $\mathbb{C}(y,f(y))$ is transcendental over $\mathbb{C}(y)$. Setting $y=\psi(x)$ is merely making an algebraic extension of each, so $\mathbb{C}(\psi(x),f(\psi(x)))$ is transcendental over $\mathbb{C}(\psi(x))$. Hence the tower $\mathbb{C}(\psi(x),f(\psi(x)))\supset \mathbb{C}(\psi(x))\supset \mathbb{C}(x)$ is transcendental, so $f(\psi(x))$ is transcendental over $\mathbb{C}(x)$. [This does seem a bit elementary for MO, so don't be surprised if it's closed.]
Aug
25
comment Asymptotics on prime divisors
I forgot to put this in my answer, so will leave it as a comment. You mention Zsigmondy's theorem. It is certainly a very interesting question to ask if all but finitely many $a(n)$ have a primitive prime divisor. I don't know the answer. But since your sequence is not a divisibility sequence, it's behavior may be somewhat different from the Zsigmondy-type results for classical and elliptic divisibility sequences.
Aug
25
answered Asymptotics on prime divisors
Aug
25
revised Separation of lattice points on the Mordell elliptic curve
Changed a plus sign to a minus sign. Added some material.
Aug
25
comment Separation of lattice points on the Mordell elliptic curve
I'd forgotten about that Danilov article. Anyway, I thought about it for a little bit and don't see an immediate way to get a gap estimate using (local) canonical height estimates. Seems like an interesting problem. My article with an elliptic curve gap principle is: A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60-100. For the curves $E_k:Y^2=X^3+k$ with $k$ 6'th power free, I prove that there is an absolute constant $C$ so that $|E_k(\mathbb{Z})|\le C^{1+rank~E_k(\mathbb{Q})}$`.
Aug
24
comment Separation of lattice points on the Mordell elliptic curve
@Lucia That would be a gap principle sort of statement. I'll have to think about it. I wrote a paper with a general gap principle for integral points on elliptic curves, but I'm not sure if it's relevant here. OTOH, if the specific question is about the $x\in\mathbb{N}$ such that there is a $y$ satisfying $0<|x^3-y^2|<\sqrt{x}$, it's not at all clear (at least to me) that the set of such $x$ is infinite. But maybe one could fix a small $\epsilon$ and take $x$ values admitting a solution to $0<|x^3-y^2|<x^{1/2+\epsilon}$, or use an upper bound of $x^{1/2}(\log x)^k$ for some fixed $k$.
Aug
23
comment Separation of lattice points on the Mordell elliptic curve
@KConrad Thanks for filling in $C$, I'd forgotten the exact value and didn't feel like rederiving, it. Actually, if you take $x$ and $y$ to be $S$-integers in a function field of genus $g$, the constant is something like $-(2g-2+\#S)$. As for the $x^3-y^2\ne0$ condition, I didn't think that it was necessary to repeat it, but I guess it doesn't hurt to put it in.
Aug
23
answered Separation of lattice points on the Mordell elliptic curve
Aug
21
comment Superelliptic Curves
If your curve is non-singular (including the point(s) at infinity), then it is an elliptic curve, or more properly, a curve of genus $1$. If it has one rational point, then it is an elliptic curve defined over $\mathbb{Q}$. If not, it is a homogeneous space for an elliptic curve over $\mathbb{Q}$. In any case, what you want to search for is "integral points on elliptic curves". You'll find a multitude of articles.
Aug
21
comment Small values of a polynomial evaluated at roots of unity
Thanks. As Doug Lind pointed out in his answer, the paper you cite has the result if the unitary set of $f$ is finite. The result for codimension 2 or greater unitary sets is in the paper: [MR3082539] Lind, Douglas; Schmidt, Klaus; Verbitskiy, Evgeny; Homoclinic points, atoral polynomials, and periodic points of algebraic $\mathbb{Z}^d$-actions. Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1060–1081.
Aug
21
comment Small values of a polynomial evaluated at roots of unity
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
comment Small values of a polynomial evaluated at roots of unity
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
comment Small values of a polynomial evaluated at roots of unity
Theorem (Erdos, Turan) Let $F(x)=\sum_{k=0}^d a_kx^k\in\mathbb{C}[x]$ with $a_0a_d\ne0$, and let $$ N(F;\alpha,\beta) = \#\bigl\{ \hbox{roots $r\in\mathbb{C}$ of $F$ with $\alpha\le\hbox{arg}(r)\le\beta$}\bigr\}. $$ Then for all $0\le \alpha<\beta\le2\pi$, $$ \left| \frac{N(F;\alpha,\beta)}{d} - \frac{\beta-\alpha}{2\pi}\right| \le \frac{16}{\sqrt{d}} \cdot \left[ \log \left( \frac{|a_0|+\cdots+|a_d|}{\sqrt{|a_0a_d|}} \right) \right]^{1/2}. $$
Aug
21
comment Small values of a polynomial evaluated at roots of unity
This is starting to look like an approach I'd tried using the equidistribution results in the following paper. Erdos, P. and Turan, P., On the distribution of roots of polynomials, Ann. Math. 51 (1950), 105-119. They prove a quantitative version of the statement that a polynomial of large degree and small length has roots whose arguments are equidistributed on the unit circle: (statement is too long for one comment, see next comment)
Aug
21
comment Small values of a polynomial evaluated at roots of unity
Thanks. A lot to absorb. I'll look at it tomorrow.