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Algebraic methods in Discrete Geometry: Ideals of Graphs and Polytopes

Subject Area Mathematics
Term from 2020 to 2024
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 454595616
 
Final Report Year 2023

Final Report Abstract

The two main results of this project concern a general framework to look for non-realizability certificates of spheres and the search for a combinatorial characterization of Cohen-Macaulay binomial edge ideals. The first result comes from the classical problem in polytope theory of deciding whether a given abstract polytopal sphere is realizable as a convex polytope. Our approach uses a variant of the classical algebraic certificates introduced by Bokowski and Sturmfels in 1989, the final polynomials. More specifically we reduce the problem of finding a realization to that of finding a positive point in a variety and try to find a polynomial with positive coefficients in the generating ideal (a positive polynomial), showing that such point does not exist. Many, if not most, of the techniques for proving non-realizability developed in the last three decades can be seen as following this framework, using more or less elaborate ways of constructing such positive polynomials. Our proposal is more straightforward as we simply use linear programming to exhaustively search for such positive polynomials in the ideal restricted to some linear subspace. Somewhat surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives, and allows us to derive new examples of non-realizable abstract polytopal spheres. We are currently working on a better implementation of our method in SageMath. The second result is the formulation and the study of a combinatorial conjecture that characterizes Cohen-Macaulay binomial edge ideals in terms of their graph. Binomial edge ideals have been introduced in 2010 as the ideals JG generated by the binomials xi yj - xj yi corresponding to the edges (i, j) of a finite simple graph G. They are a natural generalization of the ideals of 2-minors of a 2 x n generic matrix and they produced a fruitful research line. Given their combinatorial nature, many researchers have tried to interpret the algebraic properties of these ideals in terms of the associated graph. An important example is given by the minimal prime ideals of JG that turned out to be in bijection with special disconnecting sets of vertices of G, called cut sets. A challenging open problem consists in finding a combinatorial characterization of Cohen-Macaulay binomial edge ideals, which form a particularly interesting subclass of these ideals. Studying the case of bipartite graphs, we stumbled upon a characterization of the underlying graphs purely in terms of cut sets: JG is Cohen-Macaulay if and only if JG is unmixed and the collection of cut sets of G is an accessible set system. We later called these graphs accessible and realized that these conditions seemed to be common to the graphs of all Cohen-Macaulay binomial edge ideals, leading to the formulation and the study of a conjecture for all graphs. In a series of subsequent papers we proved that the conjecture holds for chordal graphs, traceable graphs and for graphs with a small number of vertices. Moreover, we recently proved a weaker version of the conjecture, showing that JG satisfies Serre’s condition (S2) if and only if G is accessible, making a further step towards the proof of the complete characterization of Cohen-Macaulay binomial edge ideals.

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