2026/06/29 by Filip Filipović, Rémi Géraud-Stewart, David Naccacheand Aleksa Veličković
#math.GM
We investigate the algebraic nature of constants arising from Chebyshev equiripple (minimax) polynomial approximation of Lp norms on [0,1]. For the Euclidean case √(1+t2), we compute equiripple solutions from degree~1 through~8 and determine exact minimal polynomials and Galois groups for degrees~1 and~2 in both absolute and relative error formulations. We find a sharp phase transition: the degree-1 constants are solvable by radicals (Galois groups C4, D4), while the degree-2 constants provably are not (Galois groups S12, S10× C2). We explain this transition by a structural dichotomy between decoupling and coupling of critical points, and extend the analysis to L3 norms, where the minimal polynomial degree jumps to~246. A general impossibility result follows from Hilbert's irreducibility theorem. We also develop a theory of piecewise equiripple approximation with jointly optimized breakpoints, proving that each doubling of the number of subintervals gains n+1 bits of accuracy at no additional arithmetic cost. These results establish a previously unobserved connection between Chebyshev approximation theory and the non-solvability of algebraic equations.