Student presentation: Algorithms for counting real roots

In the 17th century, Rene Descarte first developed ideas to count the number of positive real roots of a univariate real polynomial based on its coefficients alone. These ideas have come a long way and today provide us with useful algorithms to answer the following questions:

- Given f in R[x], how many of its roots are real? How many are positive? How many lie in [0,1]?

- Given f,g in R[x], how many points a are there such that both f(a) = 0 and g(a) > 0?

We will discuss some of these algorithms and how they can be applied to answer some of these questions.