In arithmetic geometry one encounters many important transcendental functions exhibiting interesting algebraic properties. Perhaps the most famous example of this is the complex exponential function, which is well-known to satisfy the definition of a group homomorphism. When studying these algebraic properties, a very natural question that arises is something known as the "existential closedness problem": when does an algebraic variety intersect the graph of the function in a Zariski dense set?
In this talk I will introduce the existential closedness problem, we will review what is known about it, and I will present results about a strengthening of the problem where we seek to find a point in the intersection of the algebraic variety and the graph of the function which is generic in the algebraic variety. | 
