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  
stats  profile views  18,390 
[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 powerseries 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 powerseries 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 ``TumuraHayman" 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 nonvacuous 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 $$ (1z) \, G(z) = p_0  \sum_{j=1}^\infty (p_j  p_{j1}) \, z^j $$ all the coefficients $p_j  p_{j1}$ are positive and their sum telescopes to $p_0$, so $$ \left \sum_{j=1}^\infty (p_j  p_{j1}) \, z^j \right < p_0$ $$ 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? 