Stabilisation of Iterated Toric Fibre Products

In this thesis we have analysed how the toric fibre product of certain ideals behaves when repeatedly applied to a finite number of ideals. Specifically we have investigated bounds on the degrees of generating polynomials of these iterated products. Stabilisation of the toric fibre product is said to occur when there is a uniform bound on these degrees, independent of the number of iterations. 

We have shown that stabilisation occurs when we apply the iterated toric fibre product to the vanishing ideals of the class of affine varieties that are closed under coordinate-wise multiplication. In doing so have have analysed the inverse limit of varieties that arise from these ideals. We have found new criteria for monoid algebras to be Noetherian up to the action of a monoid of endomorphisms. 

Among the investigated class of varieties for which the product stabilises are toric varieties, which occur frequently in applications. One such application, a motivating example for this research, is that of graphical models. These are statistical models defined over finite graphs. These graphs can be glued to obtain a bigger model, and this glueing corresponds to taking the toric fibre product of certain associated toric ideals. Hereby we obtain that iterating glueing of graphical models results in toric ideals that are generated in uniformly bounded degree.