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Our goal in this paper is to present a generalization of the spectral zetaregularization for general Feynman amplitudes. Our method uses complex powersof elliptic operators but involves several complex parameters in the spirit ofthe analytic renormalization by Speer, to build mathematical foundations forthe renormalization of perturbative interacting quantum field theories. Ourmain result shows that spectrally regularized Feynman amplitudes admit ananalytic continuation as meromorphic germs with linear poles in the sense ofthe works of Guo-Paycha and the second author. We also give an explicitdetermination of the affine hyperplanes supporting the poles. Our proof relieson suitable resolution of singularities of products of heat kernels to makethem smooth.  As an application of the analytic continuation result, we use a universalprojection from meromorphic germs with linear poles on holomorphic germs toconstruct renormalization maps which subtract singularities of Feynmanamplitudes of Euclidean fields. Our renormalization maps are shown to satisfyconsistency conditions previously introduced in the work ofNikolov-Todorov-Stora in the case of flat space-times.

Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups