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bio website math.brown.edu/~jhs
location Brown University Mathematics Department
age
visits member for 3 years, 10 months
seen 13 hours ago
Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

2d
comment Publishing in mathematics
Probably the answer is yes using any reasonable measure of "standard". Certainly this is the impression one gets from speaking to most editors of journals. On the other hand, there are also many more quite decent journals that are publishing many more pages of mathematics than there were ten or twenty years ago, so it doesn't necessarily follow that it's harder to find a good journal to publish an article (which I sense is at least part of what you're asking).
Oct
20
comment Who first defined quantum integers?
In the first displayed equation, presumably $m$ is supposed to be $n$.
Oct
14
revised The linear projection of projective spaces
Fixed TeX and punctuation
Oct
10
comment Why do roots of polynomials tend to have absolute value close to 1?
@IanMorris Indeed, the roots will be uniformly distributed on the unit circle, as per the result of Erdos and Turan that I described in my answer (so sorry, but this fact was not unremarked). But as you say, the heuristic of Kostlan gives a very nice, albeit rough, explanation of why it should be true. The Erdos-Turan theorem, which is not at all easy prove, gives a precise quantitative formulation.
Oct
5
awarded  Good Answer
Oct
3
awarded  Nice Answer
Oct
3
answered Why do roots of polynomials tend to have absolute value close to 1?
Oct
1
comment What is the most useful non-existing object of your field?
@ToddTrimble In fairness, it should be noted that Hellegouarch defined these curves associated to Fermat solutions and used them to prove some interesting things relating Fermat solutions to rational torsion points on elliptic curves. I believe that this was before Frey started looking at them. Not to take anything away from Frey, who made the key observation that these curves provide a link between Fermat and the modularity conjecture. Anyway, although it makes for a longer name, these non-existent curves might better called Frey-Hellegouarch curves.
Sep
30
awarded  Explainer
Sep
30
comment Is there an integral point in the group generated by an rational point?
Noam, regarding your point (1), I think it's enough that the equation for $E$ have integer coefficients. It's not necessary that it be minimal in order to deduce that $d(P)\mid d(nP)$.
Sep
30
comment Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q
I would say that Riemann-Roch is the most basic and the most powerful tool for constructing functions on curves. Further, it's proof is far more elementary (and down-to-earth) than much of the material in Chapter II of Hartshorne. So I'd suggest learning about RR and using it for problems of this sort, unless you have some good reason why a non-RR proof would be useful. (E.g., are you hoping to work on an analogous problem where RR is not available or useful? If so, it would be helpful to identify said problem.)
Sep
30
answered Is there an integral point in the group generated by an rational point?
Sep
29
comment Resultant of system with 3 polynomials and 3 variables
@JasonDeVito I think you have to homogenize, then check if the common solutions are points "at infinity". For example, do $ax+b$ and $cx+d$ have a common root. The only sensible polynomial condition to check is $ad-bc=0$. But this says that they have a common root if $a=c=0$, which is true only in the sense that if $a=c=0$, then they both vanish at the point at infinity in $\mathbb P^1$.
Sep
29
comment Resultant of system with 3 polynomials and 3 variables
@JasonDeVito Pretty much any book that has a section on "Elimination Theory" will contain this result. I first saw it in van der Waeden's "Algebra", which is a bit old-fashioned, but beautifully written. Or see Eisenbud's Commutative Algebra with a view towards Algebraic Geometry, Chapter 14, Theorem 14.1. The proof is in the exercises starting on page 318, and he refers to Mumford's Complex Projective Varieties for those who want to look up the proof.
Sep
26
comment Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
@FilippoAlbertoEdoardo It would indeed be useful for someone to give a brief description of the generalizations (I am not qualified to do so), but I did mention in the first paragraph of my answer that they exist, in particular as formulated by Tate, Beilinson, and (as you say) Bloch and Kato.
Sep
25
awarded  Enlightened
Sep
25
awarded  Nice Answer
Sep
25
revised Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
added 232 characters in body
Sep
25
comment Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
@FelipeVoloch Is the abelian variety generalization due to Tate? It may well be, I must admit that I do not know the history.
Sep
25
answered Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?