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We present recent developments on the topic of the integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the corresponding integrands. This approach has already been successfully applied and automated at one loop, and recently extended to higher loops. We describe a coherent framework based on simple concepts of algebraic geometry, such as multivariate polynomial division, which can be used in order to obtain the integrand decomposition of any amplitude at any loop order. In the one-loop case, we discuss an improved reduction algorithm, based on the application of the Laurent series expansion to the integrands, which has been implemented in the semi-numerical library Ninja. At two loops, we present the reduction of five-point amplitudes in N=4 SYM, with a unitarity-based construction of the integrand. We also describe the multi-loop divide-and-conquer approach, which can always be used to find the integrand decomposition of any Feynman graph, regardless of the form and the complexity of the integrand, with purely algebraic operations.

Multi-loop Integrand Reduction via Multivariate Polynomial Division