Critical values, resultants and norms of the Chebyshev defect

Abstract

For \(d\ge3\) let \(F_d(c)=c^d-T_d(c)\) and let \(C_d\) be its uniform norm on \([-1,1]\). We connect the critical geometry of this defect with a single cyclotomic resultant. The critical values at distinct positive critical points have distinct absolute values; a pair of monotone algebraic branches proves this without estimates on the spacing of critical points. Writing \(N=d-1\), we obtain \[disc F_d=d^d(2^N-1)P_d,\qquad P_d=\prod_{\substack{\omega^N=1\\\omega\ne1}} \bigl(2^N-(1+\omega)^N\bigr).\] This discriminant identity is unconditional. The degree and norm of \(C_d\) follow from the same critical product under irreducibility of the associated polynomial \(A_d\); the hypothesis is already known to hold for every even \(d\) with \(\gcd(v_2(d),d/2-1)=1\) (hence for \(d\equiv2\pmod4\)), as well as further families in the companion arithmetic paper. The norm formula has different multiplicities in odd and even degree. We factor \(P_d\) into cyclotomic norms and describe its prime divisors by the torsion line \(2v-u=1\), retaining the exceptional point in characteristics dividing \(N\). For odd prime-power indices, the double contact with the subtorus \(u=v^2\) determines exact valuations. A separate cyclotomic-layer argument gives the exact dyadic valuation at every index. A Gaussian lattice-sum analysis determines the square-root correction to the large resultant, while the companion density-one irreducibility theorem makes the degree and norm formulas unconditional for almost all even indices.