Decomposable form equations without the finiteness property

Abstract:

Let $K$ be a finitely generated (but not necessarily algebraic) extension field of ${\mathbb {Q}}$. Let $F({\mathbf {X}})=F(X_{1}, \dots , X_{m})$ be a form (homogeneous polynomial) in $m \ge 2$ variables with coefficients in $K$, and suppose that $F$ is decomposable (i.e., that it factorizes into linear factors over some finite extension of $K$). We say that $F$ has the finiteness property over $K$ if for every $b \in K^{*}$ (here $K^{*}$ denotes the set of non-zero elements in $K$) and for every subring $R$ of $K$ which is finitely generated over ${\mathbb {Z}}$, the equation \begin{equation*} F({\mathbf {x}})=b ~~~\text {in} ~~~~{\mathbf {x}}=(x_{1}, \dots , x_{m})\in R^{m}\end{equation*} has only finitely many solutions. This paper proves the following result: Let $F$ be a decomposable form in $m \ge 2$ variables with coefficients in $K$, which factorizes into linear factors over $K$. Let ${\mathcal {L}}$ denote a maximal set of pairwise linearly independent linear factors of $F$. If $F$ has the finiteness property over $K$, then $\#{\mathcal {L}} > 2(m-1)$.