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Questions tagged [diophantine-geometry]

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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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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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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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Recovering integrality by solving rational polynomial systems that approximate real polynomial systems

Given $n$ polynomials $g_1(x_1,\dots,x_{n}),\dots,g_{n}(x_1,\dots,x_{n})\in\mathbb R[x_1,\dots,x_{n}]$ where each of $g_1(x_1,\dots,x_{n}),\dots,g_{n}(x_1,\dots,x_{n})$ is homogeneous of degree $d$ ...
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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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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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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 ...
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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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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 $u_i$'...
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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.
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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 ...
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1answer
301 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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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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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 ...
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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/...
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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 ...
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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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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...