Critical Points and Syzygies for Feynman Integrals

We investigate a novel theoretical structure underlying the computation of integration-by-parts relations between Feynman integrals via syzygy-based methods. Building on insights from intersection theory, we analyze the large-$\epsilon$ limit of dimensional regularization on the maximal cut, showing that total derivatives vanish on the critical locus of the logarithm of the Baikov polynomial–the locus known to govern the number of master integrals. We introduce “critical syzygies” as a distinguished subset of syzygies that captures this behavior. We show that, when the critical locus is isolated, critical syzygies generate a sufficient set of total derivatives in the large-$\epsilon$ limit. We study their structure analytically at one loop and develop a numerical approach for their construction at two loops. Our results demonstrate that critical syzygies are a valuable tool for integral reduction in cutting-edge two-loop examples, offering a novel geometric perspective on integration-by-parts relations.