We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$ , then the set of all nonnegative integers n such that $\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $ , $\alpha $ and V) such that for each positive integer M,

$$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$

Let $L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $ be the $L$-series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 0} $ be a nondegenerate binary recurrence sequence. We prove that if ${ \mathcal{M} }_{E} $ is the set of $n$ such that ${a}_{n} \not = 0$ and ${ \mathcal{N} }_{E} $ is the subset of $n\in { \mathcal{M} }_{E} $ such that $\vert {a}_{n} \vert = \vert {u}_{m} \vert $ holds with some integer $m\geq 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $.

We will give explicit bounds for the number of solutions of polynomial-exponential equations. In contrast to earlier work, the bounds are independent of the coefficients of the equations, and they are of only single exponential growth in the number of coefficients.