Questions tagged [diophantine-geometry]

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Equidistribution of Hecke points and Steinitz classes

Let $K$ be a number field and $\mathcal{P}$ be a prime ideal of $K$. Let $k_{\mathcal{P}}$ be the residue field $\mathcal{O}_K/\mathcal{P}$. Consider the following construction used very often in ...
Breakfastisready's user avatar
2 votes
0 answers
111 views

The connection of Faltings height and Tate module

Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...
Richard's user avatar
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3 answers
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A rational distance problem with (possibly) multiple solutions

Background: It is not known if any point exists on the XY plane that is at rational distance from all 4 vertices of a unit square lying on the XY plane (https://mathworld.wolfram.com/...
Nandakumar R's user avatar
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0 votes
0 answers
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To tile the plane with mutually non-congruent rational triangles of equal area

We add a little to Tiling the plane with pairwise non-congruent rational triangles Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "...
Nandakumar R's user avatar
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1 vote
0 answers
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Which polygons allow partition into rational triangles?

A triangle with all side lengths rational is said to be a rational triangle. It is known - for example, Cutting the unit square into pieces with rational length sides - that the unit square allows ...
Nandakumar R's user avatar
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0 votes
1 answer
146 views

Almost Pell type equation

Consider the following Diophantine equation $$ 2x^2-Ny^2 = -1. $$ where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
Puzzled's user avatar
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8 votes
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Lattice point counts on the determinantal variety

I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$. $\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...
Breakfastisready's user avatar
1 vote
2 answers
123 views

Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$

Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials. I am interested in an upper bound for $$ N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}. $$ I assume there must be something known ...
Johnny T.'s user avatar
  • 3,527
5 votes
1 answer
110 views

How should multiplicative height on projective space interact with automorphisms?

Background on heights Consider $P = [a: b] \in \mathbb{P}^1(\mathbb{Q})$, where $a,b$ are coprime integers. We define the naive (multiplicative) height as $$H(P) = \max \{|a|, |b|\}$$ We can change ...
dummy's user avatar
  • 157
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1 answer
231 views

System of two linear Diophantine equations

Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system $$ \left\lbrace\begin{array}{l} \sum_{i=1}^nx_i = 3n; \\ \sum_{i=1}^n (2i-1)x_i = ...
Puzzled's user avatar
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7 votes
2 answers
707 views

Well known applications of Roth's theorem

Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places. It is an ...
manifold's user avatar
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0 answers
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Height on $\mathbb G^n_m$ and Néron–Tate heights

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ and let $L$ be a line bundle on $A$. Then a Néron–Tate height $\hat h_L$ can be defined by taking a model $\mathcal A$ of $A$ (over the ring ...
manifold's user avatar
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0 answers
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Integral points in smooth cubic curves

Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and $$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
var's user avatar
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0 answers
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Dyson's lemma implies index is small (in proving Roth's theorem)

I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
finiteness's user avatar
-1 votes
1 answer
128 views

Does this quadratic system admit an integral or a rational solution?

Let $a,b$ be coprime and say $0<a<b<2a$. Consider the quadratic system: $$\alpha\delta-\beta\gamma=1$$ $$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
Turbo's user avatar
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3 votes
1 answer
338 views

Rational points of bounded height on a variety

I would like to ask for some clarification on the following argument which I can not quite understand. There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
Puzzled's user avatar
  • 8,832
6 votes
2 answers
683 views

Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$

Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer. While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
ASP's user avatar
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0 answers
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Local expression of a quasi projective variety under finite morphism

I am studying the article Around the Chevalley-Weil Theorem by Zannier, Turchet and Corvaja and I am stuck in the following point: let $V$ and $W$ be two complex quasi-projective varieties and $\pi\...
cartesio's user avatar
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0 answers
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A specific Diophantine equation related to the congruent number question

Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
roydiptajit's user avatar
3 votes
0 answers
425 views

Closed immersion hitting all the $\mathbb{Q}$-points

Let $i:X\to Y$ be a closed immersion of smooth projective varieties over $\mathbb{Q}$. Assume that $Y(\mathbb{Q})$ is infinite and $X(\mathbb{Q})\to Y(\mathbb{Q})$ is surjective. Also assume that $X$ ...
guido's user avatar
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On four non-cocyclic integral points on ellipse

Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...
Turbo's user avatar
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2 votes
0 answers
216 views

Have the following summations been studied before?

Suppose $q^2-4pr<0$, and consider the set of integral points $$\mathcal Z=\{(X,Y)\in\mathbb Z^2:$$ $$px^2+qxy+ry^2+sx+ty+u=0\}$$ which lie on an ellipse. Then define $$M=\sum_{(X,Y)\in\mathcal Z}\...
Turbo's user avatar
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6 votes
1 answer
225 views

How can the number of rational points depend on the choice of height function?

Let $V/\mathbb{Q}$ be a subvariety of $\mathbb{P}^n$. There are many plausible choices of height function, some differing only by constant factors: $\max |x_i|$ (for $(x_0,x_1,\dotsc,x_n)$, $\gcd(x_1,\...
H A Helfgott's user avatar
  • 19.3k
0 votes
0 answers
280 views

Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$

I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$. Lattice point means integer coordinates and equation with integer means diophantine ...
user2548436's user avatar
1 vote
1 answer
198 views

Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application

Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil< r_1,r_2<\lceil1+\sqrt{p}\rceil$ and $$r_1\equiv mac\bmod p$$ $$r_2\equiv mbd\bmod p$$...
Turbo's user avatar
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15 votes
1 answer
439 views

Looking for a paper on transfinite diameter by David Cantor

I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while: Cantor D.: On an extension of the definition of ...
asrxiiviii's user avatar
1 vote
0 answers
143 views

Looking for a paper by Vinberg

I have been reading about the Schinzel-Zassenhauss conjecture, and have been looking for the following reference: Vinberg E.: On some number-theoretic conjectures of V. Arnold, Japanese J. Math., vol ...
asrxiiviii's user avatar
1 vote
0 answers
132 views

Clarification regarding the definition of absolute height of an algebraic number

According to page 1-2 of this paper (https://arxiv.org/abs/0906.4286), Mahler has established the inequality $$|\alpha_1 - \alpha_2| \geq H(\alpha_1)^{-(d-1)} \tag 1 $$ to be valid for all pairs of ...
asrxiiviii's user avatar
1 vote
0 answers
55 views

Diophantine bound for homogeneous system under norm conditions for solutions and system

If $b_{ij}\in[0,2^t-1]\cap\mathbb Z$ holds at every $i\in\{1,\dots,k'\}$ and $j\in\{1,\dots,k\}$ where $1\leq k'\leq k$ holds then there are $x_1,\dots,x_k\in\mathbb Z:\sum_{j=1}^kx_jb_{1i}=0\wedge\...
VS.'s user avatar
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0 votes
0 answers
132 views

Chinese remaindering to solve solvable diophantine equations

Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...
VS.'s user avatar
  • 1,806
6 votes
1 answer
440 views

Cutting the unit square into pieces with rational length sides

The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational. To cut a unit square into n (a finite ...
Nandakumar R's user avatar
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2 votes
0 answers
182 views

References for heights of algebraic or projective variety

In my research I am using the notion of a $\textbf{rational function f on a domain}\: U\subset \mathbb C^{n}$. By this I mean that the graph of $f$ is an analytic component of an affine variety $X$ ...
Espace' etale's user avatar
14 votes
0 answers
490 views

The Ax-Kochen isomorphism theorem and the continuum hypothesis

Let's recall that: (1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \...
Mohammad Golshani's user avatar
6 votes
0 answers
261 views

*Why* is Bombieri-Pila uniform?

I am about to give a couple of lectures on Bombieri-Pila/the determinant method. Bombieri-Pila gives a bound $$|C(\mathbb{Q}) \cap B \cap \mathbb{Z}^2| \ll_{d,\epsilon} N^{1/d+\epsilon}$$ for $C$ a ...
H A Helfgott's user avatar
  • 19.3k
1 vote
0 answers
118 views

Generalization of a result of Frey

In Proposition (2) in the paper [1], in below, it is proved that: Assume that $K$ is a number field and that $C$ has infinitely many points of degree $\leq d$ over $K$. (Here $d\geq 2$ is an integer)...
user131222's user avatar
1 vote
0 answers
100 views

Maximum number of bounded primitive integer points in a zero-dimensional system

Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
VS.'s user avatar
  • 1,806
1 vote
0 answers
95 views

Smallest integer lattice point by box measure in a polytope?

Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
Turbo's user avatar
  • 13.6k
5 votes
0 answers
213 views

Isomorphism classes of lattices

Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define $$ V = \{x \in \mathbb R^6 \mid A \cdot x = 0\} $$ and $$ \Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
Ramin's user avatar
  • 1,352
4 votes
0 answers
145 views

Number of nontrivial integral solutions to $f(x)=f(y)$

Let $f(x)\in\mathbb Z[x]$ be a nonconstant polynomial, and let $$g(x,y)=\frac{f(x)-f(y)}{x-y}\in\mathbb Z[x,y].$$ Let $N(B)$ denote the number of pairs of integers $(x_0,y_0)$ such that $1\le x_0,y_0\...
352506's user avatar
  • 991
2 votes
0 answers
142 views

Genus Zero Diophantine Equations and Infinite Valuations

I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem: [1] Solving genus zero diophantine ...
user112214's user avatar
-1 votes
2 answers
229 views

On distribution of size of integer points in a subspace associated to a linear diophantine equation

Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$ n/4<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space of solutions to $ACa+ADb+BCc+BDd=0$ spanned by $3\times 4$ matrix $$N=...
Turbo's user avatar
  • 13.6k
2 votes
0 answers
59 views

A question on a certain family of complete intersection varieties

Let $k$ be a field of characteristic zero. Given integers $2 \leq s \leq r < n$, define the variety $X_n$ in $P_K^n$ with coordinates $y_0, \cdots, y_n$ and $K=k(u_0, \cdots, u_n)$ ( where $u_i$'...
Sajad Salami's user avatar
9 votes
2 answers
2k views

Any simple concrete proof of Faltings theorem?

Are there simple proofs of some concrete special cases of Faltings's theorem? Any help would be appreciated.
XL _At_Here_There's user avatar
13 votes
1 answer
514 views

The number of representations of an integer as the inner product of integral lattice points

I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an ...
Ethan Splaver's user avatar
2 votes
1 answer
439 views

Counting algebraic points of bounded height

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\...
var's user avatar
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5 votes
0 answers
602 views

Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?

This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma: Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say $a_{11}x_1+\dots+...
Turbo's user avatar
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3 votes
0 answers
125 views

Finiteness of rational points on certain open subset of surfaces of general type

Let $X$ be a smooth surface of general type defined over a number field $k$. The Bombieri-Lang conjecture asserts that there is a proper Zariski-closed set $Z$ in $X$ such that for any finite ...
Sajad Salami's user avatar
2 votes
0 answers
656 views

Self intersection of theta divisor

I hope my question is not too basic here. I have a very specific setting and I am trying to not use "big theorems" to prove a well known fact algebraically. I hope someone can give me a hint. Let $J/...
Eduardo R. Duarte's user avatar
39 votes
1 answer
2k views

Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$

For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has $(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for $q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
Noam D. Elkies's user avatar
7 votes
2 answers
461 views

Obstruction to rationality of del Pezzo surfaces of degree 4

Let $X$ be a del Pezzo surface over a number field $k$. (A del Pezzo surface over $k$ is a smooth, projective, geometrically connected surface whose anti-canonical class $K_X$ is ample.) Let $d := K_X^...
R.P.'s user avatar
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