Enumerative real algebraic geometry asks questions about the real solutions to problems from enumerative geometry, when the conditions in the problem are real. This is the geometric version of the important problem of determining the real solutions to a system of polynomial equations. While this subject is fairly new (the motivating question was asked in 1984, and the first problems were studied from this perspective in the early 1990's) some basic themes are emerging.
One theme is that it seems to be always possible to arrange that the solution figures to an `honest' intersection problem are all real. When this occurs, we call the enumerative problems fully real. Another theme is that there often exist lower bounds on the number of real solutions better than trivial bounds. What is not yet clear is whether these lower bounds are sharp, or how widespread is the phenomenon of full reality.
This comprehensive article describe the current state of knowledge,
indicating these themes, and suggests lines of future research. In particular,
it compares the state of knowledge in Enumerative Real Algebraic Geometry with
what is known about real solutions to systems of sparse polynomials.