A key step in modern high energy physics scattering amplitudes computation is to express the latter in terms of a minimal set of Feynman integrals using linear relations.
In this work we present an innovative approach based on intersection theory, in order to achieve this decomposition. This allows for the direct computation of the reduction, projecting integrals appearing in the scattering amplitudes
onto an integral basis in the same fashion as vectors may be projected onto a vector basis.
Specifically, we will derive and discuss few identities between maximally cut Feynman integrals, showing their direct decomposition.
This contribution will focus on the univariate part of the story, with the multivariate generalization being discussed in a different contribution by Gasparotto and Mandal.