Characterising bases of pure difference ideals

Abstract

In this thesis, we study the basis sets of pure difference ideals, that is, ideals that are generated by differences of monic monomials. We examine the action of the hyperoctahedral group on the defining ideal of the Segre variety in the multi-dimensional case and present some striking computational results. We characterise the universal Gröbner basis for the 4-dimensional binary case of this ideal. Separately, in order to create a classification of non-toric pure difference ideals, we introduce and study the concept of a Gröbner-reversible pure difference ideal. We also outline a method for enumerating the Graver bases of some pure difference ideals that are not lattice ideals. Binomial edge ideals are binomial ideals that arise naturally from simple graphs. We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph. The maximum likelihood degree (ML degree) of an algebraic variety V is a measure of the complexity of the problem of maximum likelihood estimation for a statistical model corresponding to V. We demonstrate that the ML degree of a scaled Segre variety Vc depends on certain algebraic and geometric properties of the scaling parameter c. We give a sufficient condition for a scaled Segre variety Vc to have ML degree one and describe how the ML degree drops when c has a particular multiplicative structure. We also put forward a number of conjectures and give a closed form expression for the solutions to the likelihood equations for the simplest non-trivial case.