Hilbert’s tenth problem via additive combinatorics

Source abstract

For all infinite rings R R that are finitely generated over Z \mathbb {Z} , we show that Hilbert’s tenth problem has a negative answer. This is accomplished by constructing elliptic curves E E without rank growth in certain quadratic extensions L / K L/K . To achieve such a result unconditionally, our key innovation is to combine techniques from additive combinatorics with 2 2 -descent.