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The integrand-level methods for the reduction of scattering amplitudes arewell-established techniques, which have already proven their effectiveness inseveral applications at one-loop. In addition to the automation and refinementof tools for one-loop calculations, during the past year we observed veryinteresting progress in developing new techniques for amplitudes at two- andhigher-loops, based on similar principles. In this presentation, we review themain features of integrand-level approaches with a particular focus onalgebraic techniques, such as Laurent series expansion which we used to improvethe one-loop reduction, and multivariate polynomial division which unveils thestructure of multi-loop amplitudes.

The Integrand Reduction of One- and Two-Loop Scattering Amplitudes