Decomposable form equations without the finiteness property

Abstract

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