Effective upper bounds for the number of zeros of a linear recursive sequence (Q1084124)

Consider the recurrence sequence \(u_n=a_1 u_{n-1}+ \ldots +a_d u_{n-d}\) where the \(a_i\) are algebraic numbers. This paper gives a refinement in a well known result of Mahler concerning the set \(E=\{n;\ u_n=0\}\). For almost all prime ideals \(\mathfrak p\) of the ring of algebraic integers, there is given a number \(r_{\mathfrak p}\) and an explicit upper bound for \(\#E\) from the valuation \(v_{\mathfrak p}(u_n)\) \((0\le n<dr_{\mathfrak p})\). Then rough simplifications lead to the upper bound: \[ \# E\leq \log_2 C(6H)^{2d^ 2 d!} \] where \(C=\max (2, | u_0|,\ldots,| u_{d-1}|)\) and \(H\) is the height of the characteristic polynomial.