We establish Diophantine hardness results for the decidability of the Positivity Problem for exponential-trigonometric polynomials over computable discrete subfields of the real numbers, and for related questions. We show that any algorithm for deciding either non-negativity, eventual non-negativity, the existence of a zero, or the existence of infinitely many zeros of exponential-trigonometric polynomials over a computable discrete subfield K of the reals containing the number π can be translated into an algorithm for computing the irrationality exponents of all elements of K. As a consequence, we exhibit a computable discrete subfield K of the reals such that all of the aforementioned questions about exponential-trigonometric polynomials over K are undecidable. In particular, we provide the first example of a natural generalisation of the Continuous Skolem Problem that is provably undecidable.

Tracking of objects in cellular environments has become a vital tool in molecular cell biology. A particularly important example is single molecule tracking, which enables the study of the motion of a molecule in cellular environments by locating the molecule over time and provides quantitative information on the behavior of individual molecules in cellular environments, which were not available before through bulk studies. Here, we consider a dynamical system where the motion of an object is modeled by stochastic differential equations (SDEs), and measurements are the detected photons, emitted by the moving fluorescently labeled object, that occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to equidistant time points, which have been commonly used in the modeling of dynamical systems. The measurements are distributed according to the optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a Born and Wolf profile for an out-of-focus molecule with respect to the detector. For some special circumstances, Gaussian image models have been proposed. In this paper, we introduce a stochastic framework in which we calculate the maximum likelihood estimates of the biophysical parameters of the molecular interactions, e.g., diffusion and drift coefficients. More importantly, we develop a general framework to calculate the Cramér--Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, for the estimation of unknown parameters and use it as a benchmark in the evaluation of the standard deviation of the estimates. There exists no established method, even for Gaussian measurements, to systematically calculate the CRLB for the general motion model that we consider in this paper. We apply the developed methodology to simulated data of a molecule with linear trajectories and show that the standard deviation of the estimates matches well with the square root of the CRLB. We also show that equally sampled and Poisson distributed time points lead to significantly different Fisher information matrices.