Critical values, resultants and norms of the Chebyshev defect
Year: 2026
Type: preprint
Abstract: For the Chebyshev factorization defect \(F_d(x)=x^d-T_d(x)\), we give an unconditional discriminant identity valid for every \(d\ge3\). With \(N=d-1\), it is \(\operatorname{disc}F_d=d^d(2^N-1)P_d\), where \(P_d=\prod_{\omega^N=1,\,\omega\ne1}[2^N-(1+\omega)^N]\). The expression connects the real critical geometry of a sharp characteristic-function bound to a cyclotomic resultant. We also prove that distinct positive critical points have distinct absolute critical values. The same product yields precise algebraic-degree and absolute-norm formulas for the extremal constant \(C_d\) when the associated critical polynomial \(A_d\) is irreducible; this hypothesis is proved for extensive index families in companion arithmetic work. We factor the resultant into cyclotomic norms, characterize its prime divisors via a torsion-line condition, and determine exact dyadic and selected odd-prime valuations. The degree/norm conclusions are explicitly conditional where irreducibility has not been established. The contribution is the explicit critical-value/resultant chain, not a new definition of polynomial discriminants.
Author: Martinevski, Alexandr
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221754672
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