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Transactions of the American Mathematical Society

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The number of solutions of norm form equations

Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 317 (1990), 197-227
MSC: Primary 11D57; Secondary 11J25
MathSciNet review: 961596
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Abstract: A norm form is a form $ F({X_1}, \ldots ,{X_n})$ with rational coefficients which factors into linear forms over $ {\mathbf{C}}$ but is irreducible or a power of an irreducible form over $ {\mathbf{Q}}$. It is known that a nondegenerate norm form equation $ F({x_1}, \ldots ,{x_n}) = m$ has only finitely many solutions $ ({x_1}, \ldots ,{x_n}) \in {{\mathbf{Z}}^n}$. We derive explicit bounds for the number of solutions. When $ F$ has coefficients in $ {\mathbf{Z}}$, these bounds depend only on $ n$, $ m$ and the degree of $ F$, but are independent of the size of the coefficients of $ F$.

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