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

1d
comment Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?
@ToddTrimble The wikipedia article says "Although the word acronym is widely used to refer to any abbreviation formed from initial letters some dictionaries and usage commentators define acronym to mean..." Personally, I don't see the need to distinguish between the two, but I guess that one could come up with a situation where one might want to make the distinction. Chacun a son ...
1d
comment Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?
@Todd Is "initialization" the new term for "acronym"? :) I like it, but unfortunately initialization already has another standard meaning. (I also agree with you that on MO, one should avoid using acronyms without defining them.)
1d
comment Eigenvectors as continuous functions of matrix - diagonal perturbations
The key here, I think, is your assumption that the eigenvalue $\lambda$ is simple. ($A$ needn't be symmetric PD.) Then for $D$ in a small enough neighborhood, there is a continuous (even analytic) map $D\to\lambda(D)$ with $\lambda(0)=\lambda$ and $\lambda(D)$ an eigenvalue of $A+D$. The point is that if you take a simple root of a polynomial, then for sufficiently small perturbations of the coefficients, you get an analytic perturbation of the root. Presumably the same holds for the associated eigenvector, again since the eigenspace is one dimensional.
May
22
awarded  Civic Duty
May
19
comment A Diophantine equation with prime powers
The probability that $x$ is prime is $O(1/\log x)=O(1/n)$ and since $y\sim x$, the probability that $y$ is prime is the same. So the probability that $x$ and $y$ are both primes is $O(1/n^2)$. (Of course, I'm making some randomness and independence assumptions that can't be justified, but just to get an idea...) Summing, this suggests that there are only finitely many such solutions.
May
11
comment A property of e?
It should be noted that Clark asked a question about these sequences when $e$ is replaced by an arbitrary real numbers: mathoverflow.net/questions/204764/… Not that that makes the current question uninteresting, but I thought it would be useful to provide a link.
May
11
answered Examples of seemingly elementary problems that are hard to solve?
May
8
comment roots of reciprocal polynomials
@DaveWitteMorris I don't think that $S$ is the set of algebraic units. Not every unit is the root of a reciprocal polynomial. For example, the roots of $x^3+2x^2+1$ are units. Being reciprocal means that the reciprocals of each root is a Galois conjugate of one of the other roots.
May
8
comment roots of reciprocal polynomials
@DaveWitteMorris Actually, I think "reciprocal" is common usage, at least for people who work in Diophantine approximation. Thus it is standard to say that "Chris Smyth proved that Lehmer's conjecture is true for roots of non-reciprocal polynomials." (There's nothing wrong with "self-reciprocal", but if one is speaking of a single polynomial, the "self" seems redundant.)
May
8
awarded  Enlightened
May
8
awarded  Nice Answer
May
7
comment Reduction of tangent space of abelian variety
The definition of "good reduction" is that there is such an abelian scheme. And won't $\mathcal A$ be unique (as an abelian scheme over $R_{\mathcal P}$)? Actually, even if $A$ has bad reduction, one can specify that $\mathcal A$ be the Neron model, which will pin things down.
May
6
comment Which irrationals yield bounded sets of iterates?
Are you interested in a measure-theoretic type result, e.g., the set of $r\in(0,\infty)$ with this property has measure $0$? Or are you interested specifically in what happens when $r$ is algebraic, as in your examples? Or even when $[\mathbb{Q}(r):\mathbb{Q}]=2$, again as in your exmaples? And what happens when $r\in\mathbb{Q}$? (Since you insist that $r$ be irrational, I assume you worked out that case.)
May
6
awarded  Necromancer
May
5
comment n torsion groups of quadratic twists of elliptic curves
I can't comment on the paper, but the statement seems pretty unlikely. Even for $p^n=3$, the Galois modules $E[3]$ and $E^F[3]$ are different, so I see no reason why their first cohomology should be the same. If you want to construct an example, I'd suggest taking $E[3]\subset E(K)$.
May
5
answered n torsion groups of quadratic twists of elliptic curves
May
3
comment Computing minimal polynomials using LLL
@MarkBell If you know a priori that $\alpha$ is algebraic and you have a bound on its degree and height, then that should enable you to find the unique minimal polynomial having integer coefficients and positive leading coefficient, provided you know $\alpha$ to enough decimal places relative to the degree and height bound. OTOH, the bounds need not determine a unique polynomial with $\alpha$ as a root. For example, suppose $\alpha=\sqrt2$ (to some large precision) and your height bound is 2 and degree bound is 3. Then you might find $x^3-x^2-2x+2$ instead of $x^2-2$.
May
2
answered When can we write fundamental units explicitly
May
1
comment Computing minimal polynomials using LLL
@MarkBell You're welcome. In principle, there can't be a "correct" polynomial, since you're using a decimal approximation $\alpha$ to your algebraic number. And $\alpha$ is the root of many polynomials with integer coefficients. Of course, one wants the coefficients to be small in an appropriate sense. Anyway, it sounds as if the way PARI chooses $N$ is fairly ad hoc. So could be an interesting research project to find/prove something more precise.
May
1
comment Computing minimal polynomials using LLL
PARI-GP has a built-in function called algdep(x,d) that uses LLL to compute an integer polynomial of degree d with f(x) very small. I realize you didn't ask for a package, but the source code for PARI is public, so it should be possible to check how PARI uses LLL. And in particular, since Henri Cohen was quite active with PARI, it's likely that the implementation is a practical version of the comment you quote from his book.