AI 中文总结
该研究证明了雅可比极值的塞格（Szegő）猜想，强化了相关结论，还结合其他方法解决了至少四维球距离图的Lovász theta数的问题。
AI 中文摘要
对于参数大于$-1/2$的雅可比（Jacobi）多项式，且从端点$x=1$开始枚举相对极值，当$1\leq k\leq n$时，次数为$n+1$的第$k$个极值处的归一化模严格小于次数为$n$的第$k$个极值处的归一化模。这证明了记录在1975年第四版经典专著《正交多项式》中的塞格（Szegő）猜想，并通过移除参数的排序假设对其进行了强化。该证明将雅可比方程转化为角形式，并在相等相位处比较普吕弗（Prüfer）振幅。结合德奥利维拉·菲尔霍（de Oliveira Filho）的归约以及针对边界情形的单独二次变换论证，该结果还解决了关于至少四维球距离图的Lovász theta数的一个问题。
英文摘要
For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szegő conjecture, recorded in the 1975 fourth edition of his classic monograph Orthogonal Polynomials, and strengthens it by removing the ordering assumption on the parameters. The proof transforms the Jacobi equation to angular form and compares Prüfer amplitudes at equal phase. Combined with a reduction of de Oliveira Filho and a separate quadratic-transformation argument for the boundary case, the result also settles a question concerning the Lovász theta number of spherical distance graphs in every dimension at least four.
