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We extend the notion of generalized unitarity cuts to accommodate loop integrals with higher powers of propagators. Such integrals frequently arise in for example integration-by-parts identities, Schwinger parametrizations and Mellin-Barnes representations. The method is applied to reduction of integrals with doubled and tripled propagators and direct extract of integral coefficients at one and two loops. Our algorithm is based on degenerate multivariate residues and computational algebraic geometry.
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