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Feynman integrals obey linear relations governed by intersection numbers,which act as scalar products between vector spaces. We present a generalalgorithm for constructing multivariate intersection numbers relevant toFeynman integrals, and show for the first time how they can be used to solvethe problem of integral reduction to a basis of master integrals byprojections, and to directly derive functional equations fulfilled by thelatter. We apply it to the derivation of contiguity relations for specialfunctions admitting multi-fold integral representations, and to thedecomposition of a few Feynman integrals at one- and two-loops, as first stepstowards potential applications to generic multi-loop integrals.

Vector Space of Feynman Integrals and Multivariate Intersection Numbers