The Vertex Shift Method: Polynomial Normalization, Spectral Translation, and Dynamic Extensions Beyond the Quadratic: From Critical Points to Curved Shifts and Operator Preconditioning

We present a rigorous and comprehensive development of the Vertex Shift Method(VSM), a derivative-based polynomial normalization technique extending Loh’s quadraticmidpoint insight [1] to arbitrary-degree polynomials and time-dependent dynamicalsystems. Given a polynomial P(x) of degree n ≥2, VSM identifies a critical point φsatisfying P′(φ) = 0 and applies the shift x= y+ φ, producing a normalized form Q(y)with no linear term.We prove: (i) the linear-term elimination theorem—the shifted polynomial has zerolinear coefficient by construction; (ii) the spectral translation theorem µi = λi−φ, all rootsshift rigidly by exactly φ; (iii) companion-matrix similarity via S(φ) = exp(φD); (iv) theCurved Shift Theorem for time-varying polynomials via the Implicit Function Theorem;and (v) explicit conditioning reduction bounds. A central contribution is the identificationof VSM as a structure-aware preconditioning operator in coefficient space:eliminating a1 reduces coefficient imbalance—the principal driver of ill-conditioning incompanion matrices.Numerical experiments on eight polynomial families demonstrate conditioning improve-ments of 30×to 2.3 ×105×, 15–57% QR iteration reductions, and 3–20×root-accuracyimprovements. All results are fully reproducible via the companion Jupyter notebook.