We present two conjectures regarding the running time of computing symmetric factorizations for a Hankel matrix $\mathbf{H}$ and its inverse $\mathbf{H}^{-1}$ as $\mathbf{B}\mathbf{B}^*$ under fixed-point arithmetic. If solved, these would result in a faster-than-matrix-multiplication algorithm for solving sparse poly-conditioned linear programming problems, a fundamental problem in optimization and theoretical computer science. To justify our proposed conjectures and running times, we show weaker results of computing decompositions of the form $\mathbf{B}\mathbf{B}^* - \mathbf{C}\mathbf{C}^*$ for Hankel matrices and their inverses with the same running time. In addition, to promote our conjectures further, we discuss the connections of Hankel matrices and their symmetric factorizations to sum-of-squares (SoS) decompositions of single-variable polynomials.

翻译：我们提出两个关于在定点算术下计算Hankel矩阵$\mathbf{H}$及其逆矩阵$\mathbf{H}^{ -1}$的对称分解$\mathbf{B}\mathbf{B}^*$运行时间的猜想。若这些猜想获解，将催生一种超越矩阵乘法的算法，用于求解稀疏多项式条件线性规划问题——这是优化与理论计算机科学中的基础问题。为论证所提猜想及运行时间的合理性，我们展示了在相同运行时间内计算Hankel矩阵及其逆矩阵的$\mathbf{B}\mathbf{B}^* - \mathbf{C}\mathbf{C}^*$形式分解的较弱结果。此外，为深化对猜想的探讨，我们论述了Hankel矩阵及其对称分解与单变量多项式平方和（SoS）分解之间的联系。