Given a function $f: [a,b] \to \mathbb{R}$, if $f(a)<0$ and $f(b)>0$ and $f$ is continuous, the Intermediate Value Theorem implies that $f$ has a root in $[a,b]$. Moreover, given a value-oracle for $f$, an approximate root of $f$ can be computed using the bisection method, and the number of required evaluations is polynomial in the number of accuracy digits. The goal of this paper is to identify conditions under which this polynomiality result extends to a multi-dimensional function that satisfies the conditions of Miranda's theorem -- the natural multi-dimensional extension of the Intermediate Value Theorem. In general, finding an approximate root of $f$ might require an exponential number of evaluations even for a two-dimensional function. We show that, if $f$ is two-dimensional, and at least one component of $f$ is monotone, an approximate root of $f$ can be found using a polynomial number of evalutaions. This result has applications for computing an approximately envy-free cake-cutting among three groups.

翻译：给定函数 $f: [a,b] \to \mathbb{R}$，若 $f(a)<0$、$f(b)>0$ 且 $f$ 连续，则由介值定理可知 $f$ 在 $[a,b]$ 内存在根。进一步，若 $f$ 具有值预言机，则可通过二分法计算其近似根，所需求值次数与精度位数呈多项式关系。本文旨在识别使该多项式性结论推广至满足Miranda定理（介值定理的自然多维推广）的多维函数的条件。一般而言，即使对二维函数，寻找 $f$ 的近似根也可能需要指数次求值。我们证明：若 $f$ 为二维函数且至少有一个分量单调，则可在多项式次求值内找到 $f$ 的近似根。该结果可用于计算三个群体间的近似无嫉妒蛋糕分配问题。