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bio website math.harvard.edu/~elkies
location Harvard University, Cambridge, MA 02138
age 47
visits member for 2 years, 11 months
seen 2 hours ago
[Not sure yet what to use this space for...]

3h
comment How to prove that every polynomial in an infinite family is irreducible over Q?
Yes, if you know the Newton polygon for $(0,0)$ then you get the one for $(\infty,\infty)$ by inverting each variable and clearing the denominator. Not sure how you'd "continue using the Newton's polygon"; but Newton's method (same Newton, but not the same mathematics) works as usual: iterate the map $x \mapsto x - (P(x)/P'(x))$, getting $N$ terms of the power-series expansion for the root of $P$ in about $\log_2 N$ steps.
1d
comment How to prove that every polynomial in an infinite family is irreducible over Q?
@Peter Mueller: sorry for the misattribution; I see now that the $10^{20000}$ estimate did not come from you. Will fix in the next edit. Tyson Williams: you're welcome. I computed the series expansions for the roots using Newton's method, starting from the approximations $Y^{2/3}$ and $\pm Y^{1/2}$. These approximations, in turn, come in effect from the two sides of the Newton polygon of $p(X,Y)$ at infinity: the dominant terms are $X^5 - X^2 Y^2$ for the $Y^{2/3}$ root, and $-X^2 Y^2 + Y^3$ for the $\pm Y^{1/2}$ root.
1d
answered Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?
2d
awarded  Nice Answer
2d
comment How to prove that every polynomial in an infinite family is irreducible over Q?
Yes, they are polynomials; there are more monomials to use than power-series terms to kill (presumably this always happens unless the curve actually does have infinitely many integral points). See the new paragraphs.
2d
revised How to prove that every polynomial in an infinite family is irreducible over Q?
Exhibit polynomials, power series, etc.
Apr
16
answered How to prove that every polynomial in an infinite family is irreducible over Q?
Apr
13
awarded  Nice Answer
Apr
12
answered How can I solve a cubic equation in a finite field with characteristic 2?
Apr
8
awarded  Enlightened
Apr
8
awarded  Nice Answer
Apr
7
answered How small can a totally positive integer be?
Apr
4
answered Character table of $S_7$
Mar
28
awarded  Enlightened
Mar
28
awarded  Nice Answer
Mar
28
comment On a particular case of the ``Tumura-Hayman" theorem :
What if $p=\infty$? There was no assumption that $f$ is of finite order.
Mar
26
revised Characterization of Volumes of Lattice Cubes
Fix TeX bollix
Mar
26
comment When does a modular form satisfy a differential equation with rational coefficients?
Could it be that every modular form satisfies such an equation? Is there a counterexample, or better yet a non-vacuous necessary condition?
Mar
18
comment Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)
There cannot be zeros in $|z|<1$ or even $|z| \leq 1$ because the hypothesis $p_0 > p_1 > p_2 > \cdots > 0$ implies that in the expansion $$ (1-z) \, G(z) = p_0 - \sum_{j=1}^\infty (p_j - p_{j-1}) \, z^j $$ all the coefficients $p_j - p_{j-1}$ are positive and their sum telescopes to $p_0$, so $$ \left| \sum_{j=1}^\infty (p_j - p_{j-1}) \, z^j \right| < p_0$ $$ for all $z \neq 1$ such that $|z| \leq 1$.
Mar
16
comment Inequality regarding sum of gaussian on lattices
Interesting question. I don't remember encountering such an inequality, and don't readily see a proof or counterexample. Have you tried some examples with $n=2$ and $n=3$ to see if it seems to work numerically in all cases?