TMD2022, online, 3rd December 2022.

Title. Hilbert’s Tenth Problem and henselian valuations

Abstract. Hilbert’s Tenth Problem — originally posed for the ring $R=\mathbb{Z}$ — asked for an algorithm to determine, for any multivariable polynomial, whether or not it has a zero in $R$. The celebrated theorem of Matiyasevich in 1970 — building on work of Davis, Putnam, and Robinson — showed that no such algorithm exists for $R=\mathbb{Z}$!

I will speak about Hilbert’s Tenth Problem in the context of fields equipped with a henselian valuation.