Questions tagged [diophantine-geometry]

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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|\...
  • 393
1 vote
0 answers
52 views

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 ...
-1 votes
1 answer
125 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-\...
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4 votes
1 answer
264 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\...
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6 votes
2 answers
642 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 ...
  • 299
3 votes
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\...
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1 vote
0 answers
116 views

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 ...
3 votes
0 answers
420 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 $X$ ...
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0 votes
0 answers
72 views

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 ...
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2 votes
0 answers
211 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}\...
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6 votes
1 answer
212 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,\...
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0 votes
0 answers
202 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 ...
1 vote
1 answer
192 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 $$r_1\equiv mac\bmod p$$ $$r_2\equiv mbd\bmod p$$...
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15 votes
1 answer
423 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 ...
1 vote
0 answers
137 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 ...
1 vote
0 answers
111 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 ...
1 vote
0 answers
53 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\...
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0 votes
0 answers
103 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 ...
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5 votes
1 answer
360 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 ...
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2 votes
0 answers
168 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 $X$ ...
14 votes
0 answers
467 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}} \...
6 votes
0 answers
228 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 ...
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1 vote
0 answers
109 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)...
1 vote
0 answers
97 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^{...
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1 vote
0 answers
92 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 $A\in\mathbb Z^{m\times n}$ ...
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5 votes
0 answers
204 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 ...
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4 votes
0 answers
143 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\...
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2 votes
0 answers
136 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 ...
-1 votes
2 answers
227 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=...
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2 votes
0 answers
57 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 $u_i$'...
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.
13 votes
1 answer
460 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 ...
2 votes
1 answer
389 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(\...
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5 votes
0 answers
596 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+...
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3 votes
0 answers
119 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 ...
2 votes
0 answers
571 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/...
38 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 ...
7 votes
2 answers
409 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^...
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0 votes
1 answer
117 views

Sizes and shapes of Dedekind cuts

My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra. If we define a ...
11 votes
1 answer
793 views

Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
17 votes
1 answer
1k views

How important is Weil's decomposition theorem today?

Andre Weil's Apprenticeship of a Mathematician (p. 46) tells how he as a student realized that all of Fermat's uses of descent are unified in one principle: "If $P(x,y)$ and $Q(x,y)$ are homogeneous ...
11 votes
1 answer
644 views

Schoenberg's rational polygon problem

"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a ...
6 votes
0 answers
131 views

Diophantine approximation in $\mathbb{G}_m^r$ with approximants restricted to a finiteley generated subgroup

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and ...
6 votes
1 answer
487 views

Generalizations of de Franchis and function field Mordell

The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex ...