  This document is a contribution to the proceedings of the MathemAmplitudes
2019 conference held in December 2019 in Padova, Italy. 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 generalisation being discussed in a different contribution by
Gasparotto and Mandal.
