Birch's theorem with shifts

Autor: Sam Chow
Rok vydání: 2017
Předmět:
Zdroj: Chow, S 2017, ' Birch's theorem with shifts ', Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, vol. XVII, no. 2, pp. 449-483 . https://doi.org/10.2422/2036-2145.201510_008
ISSN: 2036-2145
0391-173X
DOI: 10.2422/2036-2145.201510_008
Popis: Let $f_1, ..., f_R$ be rational forms of degree $d \ge 2$ in $n > \sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\sigma$ is the dimension of the affine variety cut out by the condition $\mathrm{rank}(\nabla f_k)_{k=1}^R < R$. Assume that $\mathbf{f} = \mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \cdot \nabla f_k$ are linearly independent. Let $\boldsymbol{\tau} \in \mathbb{R}^R$, let $\mu$ be an irrational real number, and let $\eta$ be a positive real number. We consider the values taken by $\mathbf{f}(x_1 + \mu, ..., x_n + \mu)$ for integers $x_1, ..., x_n$. We show that these values are dense in $\mathbb{R}^R$, and prove an asymptotic formula for the number of integer solutions $\mathbf{x} \in [-P,P]^n$ to the system of inequalities $|f_k(x_1 + \mu, ..., x_n + \mu) - \tau_k| < \eta$ ($1 \le k\le R$).
Comment: Slight changes from version 1 based on comments from the referee
Databáze: OpenAIRE
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