bio | website | |
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location | Harvard University, Cambridge, MA | |
age | ||
visits | member for | 2 years, 10 months |
seen | 3 hours ago | |
stats | profile views | 998 |
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 Davenport-Lewis-Schinzel, where a more general result is proved using HIT. |
Jul 21 |
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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 case-by-case 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), 107-116 for a result better suited for this problem. |
Apr 20 |
awarded | Enthusiast |
Apr 15 |
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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 1-variable case). You understood something different, but this misundertanding is actually interesting (unless I misunderstood my own comment). |
Apr 13 |
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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 |
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What is an étale theta function?
This question is an extremely valuable contribution, regardless of how well posed it is, or what is this web-site for. Regardless of the official policy, please just keep this question, thanks. |
Feb 3 |
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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 |
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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 |
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Pairs of quadratic polynomials taking values pairs of consecutive squares
Interesting. Have you tried to do something? Do you have any guess? |
Nov 6 |
answered | Faltings height in short exact sequences |
Nov 5 |
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Is rigour just a ritual that most mathematicians wish to get rid of if they could?
Feel free to down-vote, but I could not resist: youtube.com/watch?v=aX6XMIldkRU |
Sep 5 |
awarded | Yearling |
Jul 14 |
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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 |
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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 |
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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 |
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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 |