Additive Equations

Wang Yuan
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Wang Yuan: Academia Sinica, Institute of Mathematics

Chapter Chapter 8 in Diophantine Equations and Inequalities in Algebraic Number Fields, 1991, pp 98-110 from Springer

Abstract: Abstract Let α i (1≤i≤s) be a set of nonzero integers. The form $$A(\lambda ) = \sum\limits_{{i = 1}}^{s} {{{\alpha }_{i}}\lambda _{i}^{k}}$$ is called an additive form, and the equation 8.1 $$A(\lambda ) = 0$$ its corresponding additive equation. Let R s (0) be the number of solutions of (8.1) subject to the condition: λ i ∈ P(T), 1 ≤ I ≤ s. Then R s (0) can be expressed as an integral over U n ; see §5.1. The corresponding singular series is $$\mathfrak{S}(0) = \sum\limits_{\gamma } {G(\gamma ) = \sum\limits_{a} {H(\mathfrak{a}),} }$$ where $$G(\gamma ) = \prod\limits_{{i = 1}}^{s} {{{G}_{i}}(\gamma ),}$$ $$\begin{array}{*{20}{c}} {{{G}_{i}}(\gamma ) = N{{{({{\mathfrak{a}}_{i}})}}^{{ - 1}}}\sum\limits_{{\lambda ({{\mathfrak{a}}_{i}})}} {E({{\alpha }_{i}}{{\lambda }^{k}}\gamma ),} } & {\gamma {{\alpha }_{i}}\delta \to {{\mathfrak{a}}_{i}},} & {1 \leqslant i \leqslant s,} \\ \end{array}$$ and where $$H(\mathfrak{a}) = {{\mathop{\sum }\limits_{\gamma } }^{ \star }}G(\gamma )$$ in which γ runs over a reduced system of (αδ)-1 mod δ-1.

Date: 1991
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DOI: 10.1007/978-3-642-58171-7_8

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