Let \(f(x)= P_0(x) \alpha_0^x+\cdots+ P_k(x) \alpha_k^x\) be an exponential polynomial over a field \(\mathbb{K}\) of zero characteristic. For \(i=0,\dots, k\), \(0\neq P_i\in \mathbb{K}[x]\) and \(0\neq \alpha_i\in \mathbb{K}\) such that for each pair \((i,j)\) with \(i\neq j\), \(\alpha_i/ \alpha_j\) is not a root of unity. Further, let \(\Delta= \Delta(f)= \sum_{i=0}^k (\deg P_i+1)\). The sequence \((f(n))_{n=0}^\infty\) is a nondegenerate recurrence sequence of order \(\Delta\), and every such sequence of elements of \(\mathbb{K}\) can be represented by an exponential polynomial as given above. This paper is concerned with the study of \(x\in \mathbb{Z}\) for which \(f(x)=0\), in particular with the number of such solutions. According to an old conjecture, this number is bounded by a function that depends on \(\Delta\) only. After a brief discussion of the known facts thus far, the paper proceeds by stating the main result. A partition of \(\{\alpha_0,\dots, \alpha_k\}\) into subsets \(\{\alpha_{i0},\dots, \alpha_{ik_i}\}\) \((1\leq i\leq m)\) is introduced that induces a decomposition \(f= f_1+\cdots+ f_m\), so that for \(i=1,\dots, m\), \((\alpha_{i0}:\cdots: \alpha_{ik_i}\in \mathbb{P}_{k_i} (\overline{\mathbb{Q}})\), while for \(i,u= 1,\dots, m\) with \(i\neq u\), the number \(\alpha_{i0}/ \alpha_{u0}\) either is transcendental or else algebraic with not too small a height. It is shown that for all but at most \(\exp (\Delta (5\Delta)^{5\Delta})\) solutions \(x\in \mathbb{Z}\) of \(f(x)=0\) it is true that \(f_1(x)= \cdots= f_m(x)= 0\). In particular, this result says that the conjecture can only fail when all \(\alpha_i\) belong to the algebraic closure of \(\mathbb{Q}\) in \(\mathbb{K}\). PS: After the paper was written, the conjecture was settled by the second author [\textit{W. M. Schmidt}, The zero multiplicity of linear recurrence sequences. Acta Math. 182, No. 2, 243-282 (1999)].