Given a function f: [a,b] -> 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 note 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 might require an exponential number of evaluations even for a two-dimensional function. We show that, if f is two-dimensional and satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. For any fixed dimension d, if f is a d-dimensional function that satisfies all d^2-d ``ex-diagonal'' monotonicity conditions (that is, component i of f is monotonically decreasing with respect to variable j for all i!=j), then the number of required evaluations is polynomial in the accuracy. But if f satisfies only d^2-d-2 ex-diagonal conditions, then the number of required evaluations may be exponential in the accuracy. The case of d^2-d-1 ex-diagonal conditions remains unsolved. As an example application, we show that computing approximate roots of monotone functions can be used for approximate envy-free cake-cutting.

翻译：给定函数 f: [a,b] → R，若 f(a) < 0、f(b) > 0 且 f 连续，则根据介值定理，f 在区间 [a,b] 内存在根。进一步地，若给定 f 的值预言机，可采用二分法计算其近似根，所需函数评估次数与精度位数成多项式关系。本笔记旨在确定满足 Miranda 定理（介值定理的自然多维推广）条件的多元函数中，该多项式性结论成立的适用条件。一般而言，即使对于二维函数，寻找近似根可能需要指数级评估次数。我们证明：若 f 为二维函数且满足单一单调性条件，则所需评估次数与精度成多项式关系；对于任意固定维度 d，若 f 满足所有 d²-d 个“非对角”单调性条件（即对于所有 i≠j，f 的第 i 分量关于变量 j 单调递减），则所需评估次数与精度成多项式关系；但若 f 仅满足 d²-d-2 个非对角条件，则所需评估次数可能随精度呈指数增长。d²-d-1 个非对角条件的情形尚未解决。作为应用实例，我们展示了单调函数近似根计算可用于近似无嫉妒蛋糕分割。