A note on Hadamard roots of rational functions (Q1359159)

Let \(F\) be a polynomial such that \(\sum_{k\geq 0} F(b_k) X^k\) is a rational function. Generalizing a conjecture of Pisot, the author considers the following conjecture: Suppose that all the \(b_k\) belong to a fixed field finitely generated over \({\mathbb Q}\), then prove that there exists a rational function \(\sum_{k\geq 0} c_k X^k\) such that \(F(c_k)=F(b_k)\) for all \(k\geq 0\). The paper contains a very detailed discussion of this problem and a comparison with the Hadamard quotient problem (solved by the author in 1988). Many ideas are proposed to try to get a proof of this conjecture.