This thesis deals with two problems in computational real algebraic geometry. The first is a problem of enumeration of certain flags in the real Schubert calculus. It is equivalent to the following problem in real algebraic geometry. Find a lower bound for the number of classes of real rational functions f of degree d having critical points at 2 d -3 fixed real points and satisfying f ( r ) = f ( s ) for some fixed real points r and s such that ( r, s ) contains k fixed critical points of f, 1 ≤ k < 2 d -3. We give an algorithm to determine lower bounds for all values of d and k. We also give a combinatorial interpretation of the results when k = 1, 2.
In chapter 2, we use Gröbner bases to study the set of recurrent configurations of avalanche-finite abelian sandpiles and compute the identity element and inverses in the abelian sandpile group. We then show that for sandpiles with 3 sites, the size of the reduced Gröbner basis can be made arbitrarily large. Using a result of Postnikov and Shapiro, we deduce that the same techniques can be used to find the set of G -parking functions for any directed graph G.
