Leírás
In May 2026, Martínez and Ortega-Moreno proved the Strong Polarization Conjecture, a central question in the geometry of vector systems. Their approach is based on the Euler-Jacobi vanishing theorem, an important tool in algebraic geometry. By generalizing this method we demonstrate that the polarization inequalities are consequences of a structural result regarding decompositions of the identity operator by rank-one tensors. More precisely, we develop a residue-theoretic framework for studying inverse eigenvectors of a real or complex $n \times n$ matrix $M$, that are solutions of the nonlinear equation $M\alpha=\alpha^{-1}$, where the inverse is taken coordinatewise. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions on $M$, the identity operator admits an explicit decomposition into rank-one tensors associated with the inverse eigenvectors, with explicitly computable coefficients. The proof is based on residues of rational differential forms in several complex variables.This leads to short proofs of all currently known versions of the real polarization inequalities, together with weighted and matrix-valued generalizations and further geometric and analytic applications.