The number of solutions of norm form equations

Schmidt, Wolfgang M.

doi:10.1090/S0002-9947-1990-0961596-2

The number of solutions of norm form equations
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Trans. Amer. Math. Soc. 317 (1990), 197-227 Request permission

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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