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In previous works, we devised a differential operator for evaluating typicalintegrals appearing in the Cachazo-He-Yuan (CHY) forms and in this paper wefurther streamline this method. We observe that at tree level, the number offree parameters controlling the differential operator depends solely on thenumber of external lines, after solving the constraints arising from thescattering equations. This allows us to construct a reduction matrix thatrelates the parameters of a higher-order differential operator to those of alower-order one. The reduction matrix is theory-independent and can be obtainedby solving a set of explicitly given linear conditions. The repeatedapplication of such reduction matrices eventually transforms a given tree-levelCHY-like integral to a prepared form. We also provide analytic expressions forthe parameters associated with any such prepared form at tree level. We finallygive a compact expression for the multidimensional residue for any CHY-likeintegral in terms of the reduction matrices. We adopt a dual basis projectorwhich leads to the CHY-like representation for the non-localBern-Carrasco-Johansson (BCJ) numerators at tree level in Yang-Mills theory.These BCJ numerators are efficiently computed by the improved method involvingthe reduction matrix.

BCJ Numerators from Differential Operator of Multidimensional Residue