Over the past fifty years, a trend in commutative algebra has been to explore families of homogeneous ideals, in the polynomial ring, linked to various combinatorial structures, such as graphs, hypergraphs and simplicial complexes.
In this presentation, we focus on the Eulerian ideal of a graph and its main properties. It is an ideal generated by homogeneous binomials that identify certain Eulerian subgraphs of the graph.
In the second part of the talk, we discuss one of the most important algebraic invariants of a homogeneous ideal, the Castelnuovo-Mumford regularity. We define it through the minimal graded resolution of the ideal, and recall its characterization for the Eulerian ideal.
To conclude the talk and to outline our current and future research plans, we highlight the remarkable result on the regularity of powers of a homogeneous ideal: this invariant is eventually given by a linear function.