Finiteness criteria for the solutions of a sequence of decomposable form inequalities

In this paper, we give a finiteness criterion for the solutions of the sequence of semi-$q$-decomposable form equations and inequalities, where the semi-$q$-decomposable form is factorized into a family of $q$ nonconstant homogeneous polynomials with the distributive constant not exceeding a certain number.

💡 Research Summary

The paper studies Diophantine inequalities involving a sequence of semi‑q‑decomposable forms, i.e. homogeneous polynomials that factor over an algebraic closure into a product of q non‑constant homogeneous polynomials. Let k be a number field, S a finite set of places containing all archimedean ones, and O_S, O_S^* the rings of S‑integers and S‑units. For each n≥1 a polynomial F_n∈O_S