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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.

19h
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 $x^2-2$.
1d
answered When can we write fundamental units explicitly
1d
comment Vectors:right triangle, two vertex known and a direction vector parallel to unknown point
Please read the FAQ. This sort of problem does not belong on this site.
2d
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.
2d
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.
Apr
30
answered Finite-space dynamical systems
Apr
29
answered university press specialized in math books
Apr
29
answered Elliptic Curve Multiplication
Apr
28
answered Distinct numbers in multiplication table
Apr
26
comment Average height of rational points on a curve
... and with $a$ containing some interesting arithmetic information about the curve, or more generally, variety. But as I say, that's just a guess, one would need to work out some examples to get a better feeling for what's going on.
Apr
26
comment Average height of rational points on a curve
@JosephO'Rourke The rational points on $C$ are more-or-less $P_{s,t}=(\frac{s^2-t^2}{s^2+t^2},\frac{2st}{s^2+t^2})$ with $\gcd(s,t)=1$. The height of $P_{s,t}$ is $H(P_{s,t})=s^2+t^2$. So the denominator of the limit is roughly the number of integer points in a circle of radius $\sqrt{x}$ with relatively prime coordinates, so in any case $O(x)$. The numerator is similar except we're adding $s^2+t^2$, so I think one gets $O(x^3)$, so the limit almost certainly diverges. In general, my guess is that (for reasonable situations) the growth will look like $ax^b$, with $b$ not very interesting ...
Apr
25
revised Average height of rational points on a curve
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Apr
25
answered Average height of rational points on a curve
Apr
23
comment A question on the cohomology of elliptic curves over local fields
@JakobD.Hüwer At the risk of self promotion, all of the facts in my post are, I believe, covered in my book "The Arithmetic of Elliptic Curves", especially in the proof of the Mordell-Weil theorem and descent as explained in Chapters VIII and X.
Apr
20
answered A question on the cohomology of elliptic curves over local fields
Apr
16
revised Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root
added 117 characters in body
Apr
16
answered Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root
Apr
13
answered Understanding Faltings's Theorem
Apr
12
comment Models for the moduli space $\overline{M}_{1,n}$
As a point of historical interest, this is how Neron constructed elliptic curves over $\mathbb Q$ of rank up to 10. Treating the coordinates of the 10 points as indeterminates, this gives an elliptic curve of rank 10 over the function field $\mathbb Q(T_1,\ldots,T_{20})$. Neron then proved that for "most" choices of $T_1,\ldots,T_{20}\in\mathbb{Q}$, a la Hilbert irreducibility, the 10 points remain independent.
Apr
9
awarded  Nice Answer