Some lower and upper bounds in real algebraic geometry
Abstract
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 2d–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 < 2d–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.
Degree
Ph.D.
Advisors
Gabrielov, Purdue University.
Subject Area
Mathematics
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