Workshops

## On generalized Buchi's problem

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Hilbert's Tenth Problem asks whether there is a general algorithm to determine, given any polynomial in several variables, whether there exists a zero with all coordinates in $\mathbf Z$. It was proved in the negative by Yu. Matiyasevich in 1970. In the 70's J. R. B\"uchi attempted to prove a similar statement for a system of quadric equations, and he was able to relate it to the following Diophantine problem: \medskip\noindent{\bf Conjecture (B\"uchi's square problem)}. {\it There exists an integer $M>0$ such that all $x_1,...,x_M\in\mathbf Z$ satisfying the equations $$x_1^2-2x_2^2+x_3^2=x_2^2-2x_3^2+x_4^2=\cdots=x_{M-2}^2- 2x_{M-1}^2+x_M^2=2$$ must also satisfy $x_i^2=(x+i)^2 \text{ for a fixed integer$x$and$i\in\{1,...,M\} $.}$} A generalization of Buchi's square problem asks is there a positive integer integer $M$ such that any sequence $(x_1^n,...,x_M^n)$ of $n$-th powers of integers with $n$-th difference equal to $n!$ is necessarily a sequence of $n$-th powers of successive integers. In this talk, we discuss an analogue of this problem for meromorphic functions and algebraic functions.