The Complete Extended Euclidean Scheme Is Not in Piecewise Arithmetic $\mathrm{AC}^0$

Abstract

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.