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14586
bio website math.brown.edu/~jhs
location Brown University Mathematics Department
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visits member for 4 years, 8 months
seen 35 mins ago
Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

13h
answered Examples of naturally occurring Quadratic forms or quadrics.
2d
comment summation of non-linear function
Can you format your post using LaTeX? It's very hard to read. On my screen, there's some sort of weird character after the n on each line.
2d
awarded  Necromancer
Aug
31
comment Perturbed Chebyshev polynomials
To me, the reason that the Chebyshev polynomials are so important lies in the identities $T_n((x+x^{-1})/2)=(x^n+x^{-n})/2$ and the consequence that they commute under composition. Neither of these conditions holds when you perturb, although I guess that if $a-\frac14,b-1,c,d$ are all small, you'll have small perturbations of these fundamental properties. A nice way to think of the Chebyshev polynomials is they tell you what happens to the power map $x^n$ when you identify $\mathbb R^*/(x=x^{-1})$ with $\mathbb R$ via $x\to (x+x^{-1})/2$.
Aug
31
revised What does the generating function $x/(1 - e^{-x})$ count?
added 309 characters in body
Aug
31
answered What does the generating function $x/(1 - e^{-x})$ count?
Aug
31
awarded  Nice Answer
Aug
30
comment $\mathsf{GCD}$ in arithmetic progression
Three quick comments: (1) Don't post your question if you're going to need to edit it several times in less than a minute. MO has a previewer. (2) Your question would be easier to parse if you introduce the fixed quantity $c$ first, not last. (3) There is a lot of research on so-called approximate gcd's, especially from a computational perspective; see articles by Coppersmith and Howgrave-Graham, for example.
Aug
30
answered Lower bounding the multiplicative order of 2 modulo p
Aug
27
comment Which algebraic relations are possible between algebraic conjugates?
Is that a typo? You say "If the degree is 2, the map is of the form $\frac{-x+a}{bx+1}$." I think you mean to say that "If the order is 2".
Aug
25
revised Which algebraic relations are possible between algebraic conjugates?
Added material based on a comment indicating a problem with the proof
Aug
25
comment Which algebraic relations are possible between algebraic conjugates?
@GabrielDill Sorry, you're right. The complete set of $p$-periodic points is a Galois invariant set (this is obvious), it breaks up into disjoint orbits of size $p$, so one needs to find an element of Galois that maps one of the orbits to itself, but doesn't fix the orbit. So that makes things a little trickier. The Galois group is a subgroup of a wreath product of a cyclic group $C_p$ and a permutation groups $S_N$ for an appropriate $N$. I withdraw my answer as a complete proof, but will edit it to indicate how one might proceed.
Aug
25
answered Which algebraic relations are possible between algebraic conjugates?
Aug
23
comment Rank of the Jacobian of twists of hyperelliptic curves
For elliptic curves there is a significant difference of opinion as to whether the rank of the twists $E_D(\mathbb Q)$ should be bounded or unbounded. Of course, there are also now some heuristic arguments suggesting that the rank $E(\mathbb Q)$ is bounded as one ranges over all elliptic curves. But are you suggesting, based on Petersen's article, that there exists an $E/\mathbb Q$ such that rank$(E_D(\mathbb Q))\le2$ for all $D$ (or at least for all but finitely many $D$ modulo squares)? That sounds bold.
Aug
23
comment Rank of the Jacobian of twists of hyperelliptic curves
(1) The way you've phrased the problem is confusing, since you call the Jacobian the points over $\overline{\mathbb Q}$, but then you talk about the Mordell-Weil group of $J(C)$ without specifying a field. Presumably you mean $J(C)(\mathbb Q)$. (2) The theorem you quote says that there are elliptic curves of rank at least 2, but then you ask about hyperelliptic curves of rank at most $g+1$. Do you mean at most, or at least?
Aug
20
comment Quadratic Diophantine equation in $\mathbb Z[T]$
First, it might be easier to first ask for solutions in $\mathbb C[T]$, then if those can be characterized, figure out which ones are in $\mathbb Z[T]$. Second, some motivation for your problem would be nice. (It's easy to make up lots of equations. For example, why 24?) Third, you previously asked a related, albeit easier, question (mathoverflow.net/questions/211874); it's helpful if you indicate this.
Aug
20
comment Springer GTM Reprints in China?
...in their right mind publishes books at this level with the idea of making large sums of money. One wants them widely distributed and used (and hopefully appreciated) by students and colleagues. Of course, it is also a lot of work writing a book, so I'm not going to object to receiving a couple of dollars for each one sold in the US or Europe.
Aug
20
comment Springer GTM Reprints in China?
I've had a number of my Springer books published in China just as several people have described. This is done via some sort of agreement that Springer has with some Chinese publisher, I guess. They do indeed appear to be cheaply constructed (as per Keith's comment) and contain the "for sale...only" as he says. So they are certainly being legally published in China. I get a few free copies, and there is actually a very small royalty payment, on the order of a few cents per copy, which Springer keeps half of and gives the other half to the author. OTOH, no one ...
Aug
19
answered Does the Divisor Function $\sigma(n)$ have analogues for other Fuchsian groups?
Aug
16
comment The p-adic valuation of a linear recurrence
@VesselinDimitrov Linear forms in logs ($p$-adic version) should handle $c_1A^n+c_2B^n$, as you indicate. But will linear forms in logs really handle three terms? (They may, I don't know. But for archimedean estimates, I believe that there are effective results for 3 terms, but not for 4 terms.)