Скачать или смотреть Peter Koymans: Hilbert 10 via additive combinatorics I (NTWS 251)

Abstract: Hilbert’s tenth problem asks: given a polynomial f with integer coefficients, is there an algorithm to decide whether f has an integer zero? Matiyasevich, building on earlier work of Robinson and Davis—Putnam—Robinson, showed that this problem is undecidable. He also asked what happened if Z is replaced with other rings of number-theoretic interest, for example the ring of integers O_K of a number field K.
Cornelissen, Poonen and Shlapentokh proved results of the following prototype: if there exists an elliptic curve E over K with rank equal to 1 and certain additional properties, then Hilbert’s tenth problem is undecidable for O_K. In this talk, we will give a high-level overview of our recent rank growth result on elliptic curves and show how it resolves Hilbert’s tenth problem for every number field K using a well-known reduction argument. In part II, Carlo Pagano will discuss further details and future applications of this method. This is joint work with Carlo Pagano.

Link to slides: https://drive.google.com/file/d/1eted...

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date:
Thursday, May 15, 2025 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm CST)
Friday, May 16, 2025 (1am AEST, 3am NZST)