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We present an algebraic approach to one-loop tensor integral reduction. Theintegrals are presented in terms of scalar one- to four-point functions. Thereduction is worked out explicitly until five-point functions of rank five. Thenumerical C++ package PJFry evaluates tensor coefficients in terms of a basisof scalar integrals, which is provided by an external library, e.g. QCDLoop. Weshortly describe installation and use of PJFry. Examples for numerical resultsare shown, including a special treatment for small or vanishing inversefour-point Gram determinants. An extremely efficient application of theformalism is the immediate evaluation of complete contractions of the tensorintegrals with external momenta. This leads to the problem of evaluating sumsover products of signed minors with scalar products of chords. Chords aredifferences of external momenta. These sums may be evaluated analytically in asystematic way. The final expressions for the numerical evaluation are thencompact combinations of the contributing basic scalar functions.

One-loop tensor Feynman integral reduction with signed minors