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comment 
Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$
It can be helpful to see the difference in the arithmetic side. The first proof reminds me Kummer's proof of FLT when the exponent is a regular prime (or even better, when unique factorization holds). The second proof corresponds to deducing FLT from ABC in the arithmetic side (RH formula for function fields and the truncated second main theorem for meromorphic functions are usually seen as analogues of ABC). One approach works in many cases over Z, the other remains incomplete as far as I know. 
Sep
5 
awarded  Yearling 
Aug
24 
comment 
Philosophy behind Mochizuki's work on the ABC conjecture
Apparently, this answer written in September 2012 turned out to be notsounrelated to the question. See kurims.kyotou.ac.jp/~motizuki/… 
Aug
17 
comment 
For which rings $A$ is $A\{0\}$ Diophantine over $A$?
Read the following paper by L. MoretBailly: Sur la définissabilité existentielle de la nonnullité dans les anneaux, Algebra and Number Theory. vol. 1, n° 3 (2007), 331346. 
Jul
29 
awarded  Autobiographer 
Jul
29 
revised 
Genus of a plane curve of the form $\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$
edited body 
Jul
29 
answered  Genus of a plane curve of the form $\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$ 
Jul
22 
comment 
When is the image of an integral polynomial contained in the image of another?
I should mention that the idea in this argument is a particular case of the previously mentioned paper of DavenportLewisSchinzel, where a more general result is proved using HIT. 
Jul
21 
comment 
When is the image of an integral polynomial contained in the image of another?
I think that using the suggested results will only lead to complicated issues of rationality and a lengthy casebycase analysis on the pairs in the paper of Bilu and Tichy. See Theorem 1 in: H. Davenport, D. J. Lewis, A. Schinzel, Polynomials of certain special types. Acta Arith. 9 (1964), 107116 for a result better suited for this problem. 
Apr
20 
awarded  Enthusiast 
Apr
15 
comment 
Generalization of proposition of Granville related to abc conjecture
What my comment hints is a way to prove the bound with $+1$ (using what one knows in the 1variable case). You understood something different, but this misundertanding is actually interesting (unless I misunderstood my own comment). 
Apr
13 
comment 
Generalization of proposition of Granville related to abc conjecture
Except for the "+2" (which requires finer analysis on pencils in $\mathbb{P}^2$) a bound of this sort might be proved as follows: a general linear substitution $u=at+b$, $v=ct+d$ will preserve the degree of $r,s$, the coprimality of $r,s$, and cannot increase the degree of $rad(G(r,s))$. 
Apr
4 
answered  Decidability of sum of powers exponential diophantine equation 
Feb
7 
comment 
Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$
As of today, I don't think that such a bound is known to follow from Kim's proof. Nevertheless, there is some work on computing the various objects involved in the proof, in order to get effective finiteness, see for instance 1209.0640v3 
Feb
7 
comment 
What is an étale theta function?
This question is an extremely valuable contribution, regardless of how well posed it is, or what is this website for. Regardless of the official policy, please just keep this question, thanks. 
Feb
3 
comment 
Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$
About clarifying the idea of paths: did you try reading this? people.maths.ox.ac.uk/kimm/papers/leeds.pdf 
Nov
19 
comment 
Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?
Possibly unrelated comment: If instead of strips you use discs of increasing radii then the property that you want is false but it has a reasonable fix: add proximity functions and use the first main theorem of Nevanlinna theory. If you allow different (fixed, finitely many, at least 3) choices of z then you don't need proximity functions (second main theorem). 
Nov
11 
awarded  Nice Answer 
Nov
10 
answered  Images of polynomials 
Nov
10 
comment 
Pairs of quadratic polynomials taking values pairs of consecutive squares
Interesting. Have you tried to do something? Do you have any guess? 