Generic Characteristic-Zero Equivalence Between Derivative Bézout Inversion and Multipoint Evaluation

Abstract

Let $a_1,\ldots,a_m$ be distinct elements of a field $K$, and let $Z(X)=\prod_{i=1}^m (X-a_i)$. We study the arithmetic complexity of computing the unique normalized Bezout pair $s,t$ satisfying $sZ+tZ&#39;=1$, with $°t<m$ and $°s<m-1$. The classical product-tree approach requires $O(M_K(m)\log m)$ field operations, where $M_K(m)$ denotes the cost of multiplying degree-$<m$ polynomials over $K$. Thus, even when $M_K(m)=O(m\log m)$, the resulting bound is $O(m\log^2 m)$ rather than $O(m\log m)$. Over an infinite field of characteristic zero, we prove that, in the generic rational straight-line-program model, computing all coefficients of the canonical Bezout pair is equivalent, up to an additive $O(M_K(m))$ cost, to arbitrary-node multipoint polynomial evaluation and to interpolation. The main ingredient is an explicit differential reconstruction that recovers $Z$ from $(s,t)$ in $O(M_K(m))$ arithmetic operations on a nonempty Zariski-open subset. Combining this reconstruction with automatic differentiation and transposition yields the complexity equivalence. We further transfer Strassen&#39;s lower bound for the elementary symmetric functions to the Bezout problem, obtaining an $Ω(m\log m)$ nonscalar lower bound. Hence an $O(m\log m)$ algorithm, if it exists in this model, would be asymptotically optimal. The reconstruction is genuinely characteristic-dependent: in characteristic $p$, all squarefree polynomials $X^p+cX+d$, with fixed $c\ne 0$, have the same normalized Bezout pair $(0,c^{-1})$. Therefore the all-field, all-input $O(m\log m)$ problem remains open; generically in characteristic zero, however, it is reduced to the corresponding arbitrary-node multipoint evaluation and interpolation problem.