1. Forms of differing degrees over number fields
- Author
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Christopher Frei, Manfred G. Madritsch, Institut für Analysis und Computational Number Theory (Technische Universität Graz), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and For the realization of the present paper the second author received support from the Conseil Régional de Lorraine.
- Subjects
Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Degree (graph theory) ,11G35, 11P55, 14G05 ,General Mathematics ,010102 general mathematics ,System of polynomial equations ,Singular integral ,Algebraic number field ,01 natural sciences ,Ring of integers ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Hasse principle ,0103 physical sciences ,FOS: Mathematics ,Asymptotic formula ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mathematics - 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., Comment: 23 pages; minor revision; to appear in Mathematika
- Published
- 2017