# 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 ...
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### 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 $h_F(A)$ ...
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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/...
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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 "...
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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 ...
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### 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?
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### *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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### 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)...
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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 $2^{... • 1,806 1 vote 0 answers 95 views ### Smallest integer lattice point by box measure in a polytope? Given an integer lattice$\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}$... • 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 $$\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ... • 1,352 4 votes 0 answers 145 views ### Number of nontrivial integral solutions to f(x)=f(y) Let f(x)\in\mathbb Z[x] be a nonconstant polynomial, and let$$g(x,y)=\frac{f(x)-f(y)}{x-y}\in\mathbb Z[x,y].$$Let N(B) denote the number of pairs of integers (x_0,y_0) such that 1\le x_0,y_0\... • 991 2 votes 0 answers 142 views ### Genus Zero Diophantine Equations and Infinite Valuations I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve F(X,Y)=0. I found some papers that address this problem: [1] Solving genus zero diophantine ... • 211 -1 votes 2 answers 229 views ### On distribution of size of integer points in a subspace associated to a linear diophantine equation Take A,B,C,D pairwise coprime with$$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=... • 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$u_i$'... 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. • 2,860 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 ... • 2,486 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(\... • 403 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+...
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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 ...