33
votes
Accepted
Any simple concrete proof of Faltings theorem?
Based on the OP's comment clarifying his question, I fear that the answer is no, there are no concrete special cases in which one can follow the approach of Faltings' proof that yield any significant ...
- 45.8k
20
votes
Accepted
Looking for a paper on transfinite diameter by David Cantor
Hopefully this works:
Cantor D.: On an extension of the definition of transfinite diameter and some applications
Since you said that you had "been hunting for the following paper for quite a ...
- 7,741
12
votes
Accepted
The number of representations of an integer as the inner product of integral lattice points
The asymptotic formula is true for even dimensions $k\geq 2$. We can prove this by induction on $k$, inspired by Rodrigo's observation on Eisenstein series.
The case $k=2$ is classical and addressed ...
- 99.2k
10
votes
Well known applications of Roth's theorem
If nothing else we can use Roth's theorem to generalize
Liouville's construction of transcendental numbers.
Liouville noted that numbers such as $\lambda = \sum_{k=1}^\infty 1 / 10^{k!}$
are ...
- 77.5k
8
votes
Accepted
Cutting the unit square into pieces with rational length sides
@YaakovBaruch's beautiful construction for $n=4$:
Edit by Yaakov Baruch. A partition with 8 isoceles triangles (I'm sure not minimal):
- 149k
7
votes
Accepted
Obstruction to rationality of del Pezzo surfaces of degree 4
Interesting question.
But, alas, the answer is no.
The issue is that you have missed an extra non-rationality criterion. Namely, it is possible that such a surface $X$ has $\mathrm{Br}(X) = \mathrm{...
- 20.8k
6
votes
Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
This solution works iff $q$ is a prime power, so it's not a full solution, but I guess it's better than nothing, as it is completely elementary.
If $q$ is a prime power, then $q$ must fully divide ...
- 387
6
votes
Accepted
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
The assumption that $f(X)$ and $g(X)$ are relatively prime means that there is
a positive integer $R_{f,g}=\operatorname{Resultant}(f,g)$ and polynomials $a(X)$ and $b(X)$ in $\mathbb Z[X]$ so that
$$ ...
- 45.8k
5
votes
Accepted
A rational distance problem with (possibly) multiple solutions
It is known that there are infinite sets $S$ of points on the unit circle
with rational coordinates and rational distances. So we can take any
four points of $S$, make them the vertices of $Q$, and ...
- 77.5k
5
votes
Obstruction to rationality of del Pezzo surfaces of degree 4
I will supplement Dan's nice answer by claiming that the answer is almost always no. Specifically, almost every degree 4 del Pezzo surface over the rational numbers with a rational point is non-...
- 4,155
5
votes
Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
@David Jones. Since you are not certain if the parametric solution you provide is a general solution I can confirm that it is not. There is a certain numerical solution to your equation & is given ...
- 59
5
votes
How can the number of rational points depend on the choice of height function?
Surely this behaviour can never happen, but it will be near impossible to prove this. Conjectures of Manin and others predict that there is an asymptotic formula for these functions in many cases, and ...
- 20.8k
4
votes
Any simple concrete proof of Faltings theorem?
Recently Lawrence and Venkatesh gave a proof that uses Faltings' setup but looks at variation of p-adic Galois representations in a family of algebraic varieties. Neither simpler nor a concrete ...
3
votes
System of two linear Diophantine equations
Not a complete answer, more like an extended comment.
It is possible to remove the inequalities by introducing new variables $y_i\ge 0$ according to
\begin{align}
x_n&=y_n\\
x_{n-1}-x_n&=y_{n-...
- 5,559
3
votes
Counting algebraic points of bounded height
This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. ...
- 13.7k
2
votes
Well known applications of Roth's theorem
As far as I know and as you have already pointed out above, Roth's theorem has been later extended in higher dimensions (as very often happens in mathematics) by Schmidt's subspace theorem and also by ...
- 1,119
2
votes
Accepted
Does this quadratic system admit an integral or a rational solution?
In my comments I employed Maple, which uses tools like Grobner bases to solve polynomial equations. But now I'll try to do it by hand. Let $E_1,E_2,E_3$ be the three equations. A rational solution of $...
- 37.2k
2
votes
Accepted
Rational points of bounded height on a variety
Given the situation, it's understandable that some stuff is missing from the argument.
One part that's missing is the definition of the height.
In modern language (Weil's height machine), we usually ...
- 137k
2
votes
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
The condition $f(x) | g(x), x \in \mathbb{Z}$ can be interpreted as the equation
$$\displaystyle g(x) = yf(x), x,y \in \mathbb{Z}.$$
This evidently defines an algebraic plane curve, of degree $d = \...
- 25.5k
2
votes
A rational distance problem with (possibly) multiple solutions
You should consult my joint paper with Andrew Bremner: https://www.sciencedirect.com/science/article/pii/S0022314X15002243
concernig points at rational distances from the vertices of certain geometric ...
- 791
2
votes
A rational distance problem with (possibly) multiple solutions
The rectangle with vertices $(\pm60,\pm15)$ has rational distances from the corners to both $(\pm52,0)$, because both $(15,8,17)$ and $(15,112,113)$ are Pythagorean triples.
- 39k
1
vote
Accepted
Almost Pell type equation
If you multiply this by $2$ you get $(2x)^2 - 2N y^2 = -2$, so you can solve $x^2 - 2N y^2 = -2$ and then check when $x$ is even. $x^2 - 2Ny^2 = -2$ is a generalized Pell's equation, so you can find a ...
- 2,979
1
vote
Accepted
How should multiplicative height on projective space interact with automorphisms?
The naive height is not at all intrinsic. It is just a convenient choice to work with for notational simplicity. If one is doing things properly one should be counting rational points of bounded ...
- 20.8k
1
vote
Accepted
Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application
For 3. one might use the $LLL$ algorithm to find the shortest vector to the row space of the equations $$\begin{bmatrix}bd&-ac&-p\\1&0&0\\0&1&0\end{bmatrix}\begin{bmatrix}r_1\\...
- 13.7k
1
vote
Accepted
On distribution of size of integer points in a subspace associated to a linear diophantine equation
We will assume that $n<A,B,C,D<2n$ and $\gcd(A,B)=\gcd(C,D)=1$.
For $x,y,z\in\mathbb{Q}$, we write $v(x,y,z)$ for the vector
$$\pmatrix{\frac{x}{C}&\frac{y}{C}&\frac{z}{C}}N\in\mathbb{Q}^...
- 361
1
vote
On distribution of size of integer points in a subspace associated to a linear diophantine equation
This is a partial answer in one direction. Let $k=\lceil n/6\rceil$, and let
$$A=6k+1, B=10k+1, C=9k+1, D=8k+1.
$$
The numbers $A,B,C,D$ are pairwise coprime, larger than $n$, and all their ...
- 361
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