"Some lower and upper bounds in real algebraic geometry" by Monique Edmond Azar
 

Some lower and upper bounds in real algebraic geometry

Monique Edmond Azar, Purdue University

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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