AI 中文总结
该研究证明特征零域上首一单元多项式对的完整扩展欧几里得方案等相关问题，无法由多项式规模、常深度分段算术电路计算，给出了对应下界结论。
AI 中文摘要
我们证明，对于特征零域上的首一单元多项式对，完整扩展欧几里得方案无法由Andrews和Wigderson的选择门模型下的多项式规模、常深度分段算术电路计算。实际上，该下界对输出非零欧几里得余数的完整填充列表这一更简单的任务也成立。我们表明，在非空扎里斯基开集上，可从完整欧几里得余数序列的固定坐标中恢复合适的汉克尔行列式，该关联由中间主子结式系数提供。因此，通用去除选择门后再进行常深度除法消元，会将完整余数序列的任何分段常深度算法转化为汉克尔行列式的普通常深度电路，与上述下界矛盾。我们还证明，该障碍适用于若干相关输出：其为完整多项式连分式展开和固定界主子结式系数的完整轮廓提供下界，因为这些输出均直接暴露欧几里得归约中使用的汉克尔行列式。此外，我们获得了正规化次对角帕德逼近的下界：仅正规化分母即可通过多项式多个并行帕德计算及行列式比的望远镜乘积，恢复相同的连续汉克尔行列式。因此，所有这些问题都无法由多项式规模、常深度分段算术电路计算。
英文摘要
We prove that the complete extended Euclidean scheme for pairs of monic univariate polynomials over a field of characteristic zero cannot be computed by polynomial-size, constant-depth piecewise arithmetic circuits in the select-gate model of Andrews and Wigderson. In fact, the lower bound already holds for the simpler task of outputting the complete padded list of nonzero Euclidean remainders. We show that a suitable Hankel determinant can be recovered from fixed coordinates of the complete Euclidean remainder sequence on a nonempty Zariski-open set. The connection is provided by a middle principal subresultant coefficient. A generic removal of select gates, followed by constant-depth division elimination, would therefore turn any piecewise constant-depth algorithm for the complete remainder sequence into an ordinary constant-depth circuit for Hankel determinants, contradicting the lower bound above. We also show that the same obstruction applies to several related outputs. It yields lower bounds for the complete polynomial continued-fraction expansion and for the complete profile of fixed-bound principal subresultant coefficients, since each of these outputs directly exposes the Hankel determinant used in the Euclidean reduction. In addition, we obtain a lower bound for normalized subdiagonal Pad'e approximation: even the normalized denominator alone suffices, through polynomially many parallel Pad'e computations and a telescoping product of determinantal ratios, to recover the same consecutive Hankel determinant. Consequently, none of these problems can be computed by polynomial-size, constant-depth piecewise arithmetic circuits.
CommentsAfter completion of this manuscript, I learned that Andrews and Wigderson had independently obtained the same main result. The two works were developed independently. We plan to prepare a joint final version for submission to a journal and conference
