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<-------- Over there, you'll find my website.


8h
comment Polynomial dynamical systems
When you say every function can be represented as a pair of polynomials of degree $p-1$, you mean in each variable. The degree of a polynomial in two variables is something else. Since your question is a finite one, the answer is clearly yes, so you must mean something else. Do you want a computationally efficient algorithm (perhaps polynomial in $\log p$)? Do you want an explicit criterion in terms of the coefficients of the $g_i$? Can the $g_i$ have any degree, even bigger than $p-1$?
2d
comment Questions about the Collatz conjecture-also known as the “3x+1 problem”
This question appears to be off-topic because it is about a well-known open problem.
2d
comment Cases where the number field case and the function field (with positive characteristic) are different
Diophantine approximation. Thue-Siegel-Roth fails and there are many possibilities for the approximation exponent.
Jul
23
comment function field analogy and global/absolute geometry
@WillSawin You mean Artin zeta function but, yes, it's a glaring omission.
Jul
13
comment Complex Phase Problem Relating to Binary Quadratic Forms
Binary quadratic forms usually means something else. Also $c_j^{x_j}$ is not well-defined unless $c_j = \pm 1$.
Jul
13
comment Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$
Is $8/9$ irreducible?
Jul
8
comment Equivalent binary forms
"Germany deciding to take on Brazil without using a goalkeeper." We can wish...
Jul
8
comment Degrees of maps from curves to $\mathbb P^1$
I am wrong. I was implicitly assuming that the two functions have poles at the same point.
Jul
7
comment Degrees of maps from curves to $\mathbb P^1$
$y^2=f(x)$ with $f$ of degree $2n+1$ has genus $n$. Try $n=1,2$. The function $x$ has degree $2$ and $y$ has degree $2n+1$.
Jul
7
comment Points with minimal height
Why do you mean by determine? You can plug in $\alpha=(1:0:\ldots:0)$ compute $h=H(\phi(\alpha))$, find some $h'$ such that if $H(\beta)>h'$, then $H(\phi(\beta))>h$, list all $\beta, H(\beta)\le h'$ and find the one for which $\phi(\beta)$ has minimal height.
Jul
7
comment Arithmetic and moduli spaces of higher genus curves
Faltings's proof of the Mordell conjecture takes place in the moduli space of abelian varieties.
Jul
7
comment Degrees of maps from curves to $\mathbb P^1$
If the curve is hyperelliptic of genus $g$ and $a=2$, then the lowest odd degree of a map is $b=2g+1$ and your bound $ab-a-b=2g-1$ for $c$ is optimal. This follows from Clifford's theorem, which might help in general too.
Jul
5
comment Non-hyperelliptic curves of genus at least two
See also mathoverflow.net/questions/138581/…
Jul
3
comment Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?
The identity $L(E,\bar{s})=\overline{L(E,s)}$, which is clear for $\Re(s)>3/2$, analytically continues.
Jul
3
comment Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?
1 follows from the L-function being real on the real axis. Maybe 2 likewise follows from the functional equation.
Jul
3
comment The equation $f(x)=f(a^n)^k$ always has a solution in $\mathbb{Q}$
If it is an olympiad problem, it shouldn't be on MO.
Jul
3
comment Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?
1 and 2 are known. 3 and 4 are not right unless $r=0$, because the height pairing doesn't take rational values. If $r=0$, then maybe 3 and 4 are known by Kolyvagin, but I am not sure.
Jul
2
awarded  Curious
Jun
27
comment Fixed points of $x\mapsto 2^{2^{2^{2^x}}} \mod p$
Is there any rationale for the size of the tower? Do you know the answer with fewer exponentiations?
Jun
25
comment Transcendental numbers in the p-adic rationals $\mathbb Q_p$
It does not make sense to ask if $\pi$ is in $\mathbb{Q}_p$. Even in situations where you are tempted to say a certain real number is in $\mathbb{Q}_p$ (e.g. $\sum p^n/n!$ converges $p$-adically and defines a transcendental number) it is not correct as there is no canonical way to embed $\mathbb{Q}_p$ in $\mathbb{C}$. In particular, my parenthetical number is not $e^p$ in any meaningful way. This question is based on a misunderstanding of the definitions and should be closed.