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bio website math.harvard.edu/~elkies
location Harvard University, Cambridge, MA 02138
age 48
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3h
comment Question regarding a theorem of Erdos and Renyi on $B_2(g)$ sequence
Looks like this is just counting in two ways solutions of $k=s_1+s_2$ with $s_1 \leq s_2 \leq n$ and $k \leq 2n$ to get $S(n)^2 \leq 4gn$.
2d
awarded  Enlightened
Jan
25
comment Identity for Power Series and Binomial Coefficients
You're welcome. For other $j$ there doesn't seem to be such a nice formula, except in the roughly-complementary case of $j=k-1$. Otherwise the power-series coefficients, written as polynomials in $N$, are generally irreducible once you remove the factor of $N$. (For $j=1$ they factor completely, which soon led me to surmise the formula above.)
Jan
25
comment Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Thanks. Basically I just used the routine lindep in gp. Once one has guessed such an identity numerically it's usually not hard to prove. Here $f(n)$ is odd and 7-antiperiodic, so $\sum_{n=1}^\infty f(n)/n = 0$ comes down to a linear relation among the cosecants of multiples of $\pi/7$, namely $$ \frac1{\sin \pi/7} = \frac1{\sin 2\pi/7} + \frac1{\sin 3\pi/7}. $$
Jan
25
answered Identity for Power Series and Binomial Coefficients
Jan
25
comment Identity for Power Series and Binomial Coefficients
What's the connection with representation theory?
Jan
25
awarded  Nice Answer
Jan
24
answered Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Jan
20
comment Number of critical points of smooth functions on $S^1$
There is a generalization in $T^1$, though: if $u$ is orthogonal to $\cos n\theta$ and $\sin n\theta$ for each positive $n < d$ then $u$ has at least $2d$ critical points. (Your question is the case $d=2$; both John Pardon's proof and the one I gave readily generalize to arbitrary $d$.)
Jan
20
comment Number of critical points of smooth functions on $S^1$
Sorry, I don't see a direct generalization to higher-dimensional spheres.
Jan
20
revised Number of critical points of smooth functions on $S^1$
corrected typo in second display (second integral)
Jan
20
answered Number of critical points of smooth functions on $S^1$
Jan
13
awarded  Nice Answer
Jan
12
comment Reference for hyperelliptic curves
Thanks. I must have learned this from Joe Harris.
Jan
11
comment Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
(About the older comment: sure, but at some point it becomes infeasible to factor bivariate polynomials whose degree grows with $km$ while it's still possible to compute resultants of degrees $k$ and $m$.)
Jan
11
comment Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
That's covered by the $(u',v')$ test I gave. When $m$ is even you can't take $u,v$ to be the $X,Y$ of the classical Weierstrass equation because then the map $(u,v) \mapsto (u^k,v^m)$ is not generically $1:1$ (try $(u',v') = (u,-v)$). But $(X,Y+1)$ works. (Also for $Y^2=X^3+2$ if $3 \mid k$ you must tweak $u$ to avoid triplication on the $X$ side.)
Jan
10
comment Reference for hyperelliptic curves
$\ldots$, $\kappa$ is a $2:1$ map to a rational normal curve, and if $\lambda$ preserves that curve $\kappa(C)$ then it comes from a fractional linear transformation of ${\bf P}^1$, etc.
Jan
10
comment Reference for hyperelliptic curves
More generally: A curve $C$ of genus $g>1$ has a canonical map $\kappa: C \rightarrow {\bf P}^{g-1}$. Since it's canonical, for any automorphism $\alpha: C \rightarrow C$ the composite map $\kappa \alpha: C \rightarrow {\bf P}^{g-1}$ must be the same as $\kappa$ up to some linear automorphism $\lambda$ of ${\bf P}^{g-1}$, so $\kappa\alpha = \lambda\kappa$. For example, the automorphisms of a plane quartic are exactly the linear automorphisms of the plane that preserve the quartic. In our hyperelliptic setting$\ldots$ [cont'd because of 600-character limit]
Jan
10
comment Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
You're welcome. I think the resultant is right. I just edited to add the gp code. I also checked that a few multiples of $(-1,1)$ yield rational multiples whose coordinates are the expected $11$th and $7$th powers.
Jan
10
revised Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
add requirement that u,v have no zero or pole at the test point, and a line of gp code; fix typos;