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Integration-by-parts identities between loop integrals arise from thevanishing integration of total derivatives in dimensional regularization.Generic choices of total derivatives in the Baikov or parametricrepresentations lead to identities which involve dimension shifts. Thesedimension shifts can be avoided by imposing a certain constraint on the totalderivatives. The solutions of this constraint turn out to be a specific type ofsyzygies which correspond to logarithmic vector fields along the Gramdeterminant formed of the independent external and loop momenta. We present anexplicit generating set of solutions in Baikov representation, valid for anynumber of loops and external momenta, obtained from the Laplace expansion ofthe Gram determinant. We provide a rigorous mathematical proof that this set ofsolutions is complete. This proof relates the logarithmic vector fields inquestion to ideals of submaximal minors of the Gram matrix and makes use ofclassical resolutions of such ideals.

Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals