Straightening Out the IBP Equations

For over four decades, IBP identities have been a cornerstone in the computation of multi-loop amplitudes. Yet, remarkably, our understanding of their underlying properties and general solutions remains very limited. To date, all approaches to solving them have essentially relied on increasingly sophisticated forms of brute force. In this talk, we will present ongoing work that, for the first time, successfully diagonalizes the IBP equations, casting them into a form where analytic solutions become feasible. The diagonalized equations become indispensable when propagator powers are treated as abstract variables (i.e. Mellin representation) or when very high powers are involved. As a byproduct of this progress, we can quantify the complexity of various methods for their solving and have developed a new ``numeric” approach that offers advantages over existing ones. Our work — and this talk — focuses on the recurrence-relation aspect of the IBP identities and, as we will see, can be applied much more broadly. Examples are the Fibonacci numbers and Gauss’s hypergeometric contiguous relations.