Extended spectral breakdown theory

Spectral break theory, as presented in this article, offers an algebraic-geometric framework unifying the concepts of renormalization and decoherence in the context of complex polynomials with multiplicity at least 2. The study focuses on transforming an unstable complex polynomial P , possessing at least one multiple root, into a simple polynomial Q , devoid of multiple or complex roots, through the composite operator . The renormalization operator R reduces root multiplicities to 1, while the decoherence operator D projects non-real roots onto their real parts. A spectral metric is defined to quantify the distance between the roots of P and Q, proving that ( d(P,Q) > 0 ) for any non-trivial P , establishing qualitative spectral non-proximity. Local continuity of the operators R and D is demonstrated on strata with fixed multiplicities in a non-compact space, leveraging the Weierstrass preparation theorem. Two indicators are introduced: e, measuring the impact of decoherence, and t, combining the spectral distance and e to assess the net effect of the spectral break. Examples illustrate that t reflects the dominance of either R or D . The study concludes with perspectives for extending the theory to non-linear polynomials and enhancing numerical algorithms for practical applications