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Jul
14
comment how to find cubic polynomial that an unknown subset of a set of integers satisfies
Is there a particular reason related to your application for suspecting that the coefficients are integers (rather than any integer-valued cubic, which can introduce some small denominators)? It would be interesting to know where does the problem come from (only if this is not top-secret information, of course).
Apr
28
comment How many solutions to $2^a + 3^b = 2^c + 3^d$?
Let me complement this answer, for the first step: to show that 2^a-2^c and 3^b-3^d are nearly "pure powers" you can use a lemma of Hensel (sometimes known as lifting the exponent lemma) instead of linear forms in log's. For instance, say that c<a, then c is the 2-adic valuation of (3^b - 3^d) and you can bound it with the lemma (linear forms in p-adic logarithms also work here).
Apr
28
comment Possible counterexample to a theorem assuming Lang's conjecture
At some point, I did check it (including Vojta's paper) and indeed, one just want evaluate the quadrics az^2+bz+c at 8 given points. In this paper, the 8 points are the squares of the alphas.
Apr
23
comment Units of $\mathbf Z[X,Y]/(P(X,Y))$
One can give an upper bound for the rank in terms of the places where poles are allowed (poles in the sense of valuations) by adapting the classical proof for rings of dimension 1. To deal with poles at geometric places, one can use as inspiration the analysis in J. Denef, The Diophantine problem for polynomial rings of positive characteristic. Logic Colloquium '78 (Mons, 1978), pp. 131–145
Apr
19
answered l-functions of calabi-yau varieties
Sep
18
comment Is the field of constructible numbers known to be decidable?
Write to Videla and ask him. He surely knows the state of the art and he will be happy to answer; he is a very nice person. As I understand, the problem is open and there is a group of people (including Videla) looking into it.
Sep
5
awarded  Yearling
Oct
23
awarded  Good Answer
Sep
17
awarded  Commentator
Sep
17
comment Philosophy behind Mochizuki's work on the ABC conjecture
(cont.) The table in page 27 of IUTT-I gives an idea of what are the roles played by some of the main objects introduced by Mochizuki (and as VD pointed out, the hyperbolic curve "is" the number field, not the elliptic curve). Anybody can read this directly from the paper, but the only reason why I am mentioning it is the following: I was very curious about the papers (as everybody else), but the first couple pages seemed very intimidating. However, after spending some time with the papers on Frobenioids then the introduction of IUTT-I became readable after all. I hope this suggestion helps!
Sep
17
comment Philosophy behind Mochizuki's work on the ABC conjecture
For people wanting to known more details without having to read all the 500 pages: the answer provided by VD is a nice survey of the first 16 pages of IUTT-I (avoiding technicalities of how the several types of Hodge theaters are actually constructed, or what are the prime-strips). For the interested reader, the first 27 pages of IUTT-I indeed give a very good introduction. However, it is better to get used with the language of Frobenioids FIRST, otherwise the exposition can be intimidating. Unfortunately, it does not hint on the actual "source of inequality" (I mean, not beyond analogies).
Sep
17
awarded  Nice Answer
Sep
17
awarded  Nice Answer
Sep
17
awarded  Critic
Sep
17
awarded  Supporter
Sep
17
awarded  Editor
Sep
16
comment undergraduate logic textbook
Very nice suggestion!
Sep
16
comment Philosophy behind Mochizuki's work on the ABC conjecture
@VD: I think one should read more carefully the hypothesis. Also, I would not be surprised if the final Diophatine statement is not 100% correct as stated and needs to be refined - it is such a long and complicated work!. However, I think that the whole point is the technique: if it is correct$-\epsilon$ then people will make it work at some point. I do not remember the solution of a BIG problem that was 100% correct the first time it was released (perhaps I am exaggerating a little bit).
Sep
16
comment Philosophy behind Mochizuki's work on the ABC conjecture
If you want to apply the theorem 1.10 with initial theta data having F=Q then you have a problem: F must contain i (square root of -1). If E was already semi-stable over Q then I guess that nothing happens, but otherwise the height of E gets smaller. Also, you need semi-stable reduction of E over F as part of the conditions. For example, for the Frey curve associated to an ABC triple to be semi-stable over Q, you need that the ABC triple (a,b,a+b) must be primitive (no common factor) and 16 must divide ab(a+b) (perhaps not 16...). This slightly reduces the list "too-good-to-be-true" examples.
Sep
11
comment Beautiful theorems with short proof
Zagier's paper "values of zeta functions and their applications" has a nice short proof of $\zeta(2)=\pi^2/6$ due to Calabi.