Integrand reduction at NLO and beyond

The integrand-level methods for the reduction of scattering amplitudes are powerful techniques for the analysis and the computation of loop integrals, which have already been successfully applied and automated at one-loop. Moreover, some very interesting progress has recently been made towards the higher-loop extension of such techniques. In this presentation, we review the basics principles of integrand-reduction methods within a coherent framework we developed, which can be applied to any integrand at any number of loops and is based on simple concepts of algebraic geometry, such as multivariate polynomial division. We particularly focus on semianalytic and algebraic techniques, such as the Laurent series expansion which we exploited to improve the one-loop reduction with the library NINJA, and the multi-loop divide-and-conquer approach which can always be used to find the integrand decomposition of any Feynman graph in a finite number of algebraic operations.