It is well-known, through the combined efforts of various number theorists, that exhibiting elliptic curves of rank that is positive and constant under a (sufficiently general) quadratic extension of number fields can be used to prove Hilbert 10th problem for all finitely generated infinite commutative rings. Until very recently, the existence of such curves was established only under BSD, through a descent argument in a full quadratic twist family: without BSD one loses unconditional control on the existence of rational point. In joint work with Peter Koymans we took the opposite approach. We work with a polynomial-twist sub-family of curves, almost all of them coming with a marked non-torsion point. Now the challenge is to control the descent process. This is done by judiciously specializing in the family by means of additive combinatorics in such a way to obtain sufficient control on the relevant Selmer group governing the change of rank in the extension. We deduce that Hilbert 10 is undecidable over any finitely generated infinite commutative ring. In this talk I will explain in detail this method of constructing the desired elliptic curves with positive and unchanged rank.