关于最优多项式逼近常数的代数复杂性
On the Algebraic Complexity of Optimal Polynomial Approximation Constants
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中文总结 AI 辅助
研究[0,1]上\(L_p\)范数切比雪夫等波纹多项式逼近常数的代数性质，通过计算欧几里得情形下不同次数的解及相关群，发现相变，扩展到\(L_3\)范数，还发展分段逼近理论，建立了与代数方程不可解性的联系。
中文摘要 AI 辅助
我们研究了在[0,1]上\(L_p\)范数的切比雪夫等波纹（极小极大）多项式逼近中出现的常数的代数性质。对于欧几里得情形\(\sqrt{1 + t^2}\)，我们计算了从1次到8次的等波纹解，并确定了1次和2次在绝对和相对误差公式下的精确极小多项式和伽罗瓦群。我们发现了一个明显的相变：1次常数可由根式求解（伽罗瓦群\(C_4\)，\(D_4\)），而2次常数则不可（伽罗瓦群\(S_{12}\)，\(S_{10}\times C_2\)）。我们通过临界点的解耦和耦合之间的结构二分法来解释这种转变，并将分析扩展到\(L_3\)范数，其极小多项式次数跃升至246。从希尔伯特不可约性定理得出一个一般的不可能性结果。我们还发展了具有联合优化断点的分段等波纹逼近理论，证明了子区间数量每翻倍可在不增加算术成本的情况下提高\(n + 1\)位精度。这些结果建立了切比雪夫逼近理论与代数方程不可解性之间以前未观察到的联系。
英文摘要
We investigate the algebraic nature of constants arising from Chebyshev equiripple (minimax) polynomial approximation of $L_p$ norms on $[0,1]$. For the Euclidean case $\sqrt{1+t^2}$, 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 $C_4$, $D_4$), while the degree-2 constants provably are not (Galois groups $S_{12}$, $S_{10}\times C_2$). We explain this transition by a structural dichotomy between decoupling and coupling of critical points, and extend the analysis to $L_3$ 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.
