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We prove a result on the representation of squares by second degreepolynomials in the field of $p$-adic meromorphic functions in order to solvepositively B\"uchi's $n$ squares problem in this field (that is, the problem ofthe existence of a constant $M$ such that any sequence $(x_n^2)$ of $M$ - notall constant - squares whose second difference is the constant sequence $(2)$satisfies $x_n^2=(x+n)^2$ for some $x$). We prove (based on works by Vojta) ananalogous result for function fields of characteristic zero, and under aConjecture by Bombieri, an analogous result for number fields. Using anargument by B\"uchi, we show how the obtained results improve some theoremsabout undecidability for the field of $p$-adic meromorphic functions and thering of $p$-adic entire functions.

Representation of squares by monic second degree polynomials in the field of 𝑝-adic meromorphic functions