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18h
comment What area of maths have I reinvented?
You might enjoy Hardy's book Divergent Series, which gives many methods for making sense of series that diverge.
21h
answered Uniform bounds on the number of integer points on a family of elliptic curves
Feb
8
comment Grothendieck, A Place to Begin
@StevenLandsburg Good point. I learned the non-scheme version from Shafarevich's book, which is (I thought) quite well written. Mumford, of course, is also a good choice. Best of all, I'd say, is to take a graduate algebraic geometry course a couple of times from different teachers who present it from different viewpoints. My own experience was first with Gieseker (quite geometric), second with Hironaka (very algebraic), with the summer between spent reading Hartshorne and doing lots of the problems with a group of other grad students.
Feb
8
comment Grothendieck, A Place to Begin
Not sure this is appropriate for MathOverflow, probably more appropriate on MathStackExchange, so don't be surprised if it's closed. But I'd suggest (1) learn some commutative algebra at the level of, say, Atiyah-Macdonald's or Eisenbud's books and then (2) read Hartshorne's Algebraic Geometry, which is a standard introduction to Grothendieck-style algebraic geometry.
Feb
6
comment I am looking for a general solution for when $n$ and a rational function $f (n)$ are both integers
Instead of general techniques, wouldn't one just use high school divison of polynomials to write $\frac{r^3}{r-1}=r^2+r+1+\frac{1}{r-1}$, so $\frac{r^3}{r-1}\in\mathbb Z$ if and only if $\frac{1}{r-1}\in\mathbb Z$, which is clearly true if and only if $r\in\{0,2\}$? Yes, I know this is essentially Rene's comment, but it doesn't even require the "cleverness" of writing $r^3$ as $r^3-1+1$. (In any case, this seems too elementary for MO.)
Feb
6
comment A simple question about ordinary diffential equations of first order
For your particular equation, the polynomial $T^5+T^4+1=0$ has five (complex) roots $t_1,\ldots,t_5$, and then the solutions to your equation are $y(x)=t_i x + c_i$. If you make the equation a bit more complicated, things aren't as easy, but as already noted, this question really belongs on MathStackExchange.
Feb
4
comment Degree of a rational Function
Actually, it's neither, unless you assume that $f(x)$ and $g(x)$ have no common non-constant factors. With that assumption, at least in algebraic geometry, $\deg f(x)/g(x)$ is $\max\{\deg f,\deg g\}$, since that is the degree of $f/g$ as a self-map of the projective line $\mathbb P^1\to\mathbb P^1$. But as was noted, this sort of question is better suited to MathStackExchange, rather than MathOverflow.
Jan
31
comment Galois cohomology of a non-abelian group over a function field
Another typo is "over $X$" should be "over $k$".
Jan
30
awarded  Necromancer
Jan
28
awarded  Nice Answer
Jan
28
awarded  algebraic-number-theory
Jan
28
comment Upper bound on answer for Pell equation
@Lucia I stated Siegel's theorem precisely and then prefaced the rest of the post with "at least approximately." The poster is looking at experimental data, so rather than guessing a bound of $p^{\sqrt p}$, he'd be better off first trying to estimate the growth rate of the upper bound for $\log|x_0|$, which would hide some of the lower order phenomena. Then, as you say, one should be able to guess $\gg\ll p^{1/2\pm\epsilon}$. Following Siegel blindly and pretending lower order terms don't exist gives what I wrote; I made no claim it was accurate. Thanks for being more precise.
Jan
27
answered Upper bound on answer for Pell equation
Jan
26
comment estimate sum of $(\log \log p)^2/p$
Yes, of course.
Jan
26
comment estimate sum of $(\log \log p)^2/p$
Actually, the answer to that earlier question seems to yield (at least) something like $$\sum_{p\le x} \frac{(\log\log p)^k}{p} = \frac{1}{k+1}(\log\log p)^{k+1}+O(\log\log p)^k.$$
Jan
26
comment estimate sum of $(\log \log p)^2/p$
Surely you should provide a reference your earlier question about $\sum_{p\le x} \log\log p/p$ (mathoverflow.net/questions/226784/…), to which GH from MO provided a detailed answer. Does the method used there yield anything? If not, what goes wrong?
Jan
25
comment Quadrics cutting out a polygon
Please proofread and fix this before it gets closed. First, presumably $i=i+1\bmod 5$ should be something like $i\ne j\pm1\bmod 5$, since your previous condition says that "adjacent" $l_i$ do intersect. So if I understand correctly, your five lines form a generalized pentagon. Second, the union of five lines is not an elliptic curve, since an elliptic curve is a smooth genus 1 curve (with specified basepoint). Possibly you're using some generalized definition of elliptic curve. OTOH, possibly this is irrelevant to your problem, in which case you can omit and and just say the union is a LCI.
Jan
24
awarded  Revival
Jan
22
awarded  Nice Answer
Jan
22
comment Dynamical Mordell-Lang on Kahler manifolds?
I guess you want to replace the subvariety $V$ by a sub-Kahler manifold. I'll reveal my ignorance and ask if non-projective K3s admit interesting endomorphisms that preserve the Kahler structure. If so, then this sounds like a natural question. BTW, you should add the "arithmetic-dynamics" tag to your question.