bio  website  ma.utexas.edu/users/voloch 

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< Over there, you'll find my website.
8h

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Polynomial dynamical systems
When you say every function can be represented as a pair of polynomials of degree $p1$, 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 $p1$? 
2d

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Questions about the Collatz conjecturealso known as the “3x+1 problem”
This question appears to be offtopic because it is about a wellknown open problem. 
2d

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Cases where the number field case and the function field (with positive characteristic) are different
Diophantine approximation. ThueSiegelRoth fails and there are many possibilities for the approximation exponent. 
Jul 23 
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function field analogy and global/absolute geometry
@WillSawin You mean Artin zeta function but, yes, it's a glaring omission. 
Jul 13 
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Complex Phase Problem Relating to Binary Quadratic Forms
Binary quadratic forms usually means something else. Also $c_j^{x_j}$ is not welldefined unless $c_j = \pm 1$. 
Jul 13 
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Goldbachtype problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$
Is $8/9$ irreducible? 
Jul 8 
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Equivalent binary forms
"Germany deciding to take on Brazil without using a goalkeeper." We can wish... 
Jul 8 
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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 
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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 
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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 
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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 
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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 $abab=2g1$ for $c$ is optimal. This follows from Clifford's theorem, which might help in general too. 
Jul 5 
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Nonhyperelliptic curves of genus at least two
See also mathoverflow.net/questions/138581/… 
Jul 3 
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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 
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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 Lfunction being real on the real axis. Maybe 2 likewise follows from the functional equation. 
Jul 3 
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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 
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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 
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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 
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Transcendental numbers in the padic 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. 