Forms of differing degrees over number fields
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In this paper, a system of polynomials in many variables over the ring of integers of a number field was considered and an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes was given.Abstract:
Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree.
This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.read more
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Rational points and zero-cycles on rationally connected varieties over number fields
TL;DR: The authors report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties.
Schmidt rank and algebraic closure
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TL;DR: In this paper , the authors gave polynomial bounds for counting integer points when k is a number, a function, or a function in terms of rk ¯ k (Q) where ¯ k is the algebraic closure of k .
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Solvable points on smooth projective varieties
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Forms over number fields and weak approximation
TL;DR: In this article, the Hardy-Littlewood Circle Method is used to estimate the number of integral points on affine varieties in homogeneously expanding regions, provided s exceeds some function of the degree and codimension of X.