In this work, a functional variant of the polynomial analogue of the classical Gandy's fixed point theorem is obtained. Sufficient conditions have been found to ensure that the complexity of the recursive function does not go beyond the polynomial. In 2021, we proved a polynomial analogue of the classical Gandy's fixed point theorem. This became an important impetus for the construction of p-complete programming languages. And such a language was first built by us in 2022. The main result of that work was: a solution of the problem P=L. Next are the followers of the works on building a new high-level language and the idea of building a general programming methodology. But there was one gap in our research: classes of recursive functions whose complexity was polynomial were not described. In this work we found sufficient conditions for such functions. In many ways, the main ideas of this work are similar to the ideas that we used in the proof of the polynomial analogue of Gandy's fixed point theorem. But there are also striking differences. Functions, as such, differ quite strongly from predicates precisely in the multitude of their values. If a predicate is either true or false, then a function can generally take on a variety of values. Moreover, even if there are not many values, but there is recursion and simple multiplication, then powers and factorials may arise during the calculations, which, of course, can violate the polynomial computational complexity of this function. Therefore, finding these restrictions on recursive functions that would be soft enough for the class of functions to be large, and at the same time tough enough not to go beyond polynomiality, has been a problem for us for the last 3 years, after the proof of the polynomial analogue Gandy's fixed point theorem in the case of predicate extensions.

翻译：本文获得了经典甘迪不动点定理多项式模拟的函数式变体。我们找到了确保递归函数复杂度不超越多项式级的充分条件。2021年，我们证明了经典甘迪不动点定理的多项式模拟，这为构建p完全编程语言提供了重要推动力。此类语言于2022年由我们首次实现，该工作的主要成果是：解决了P=L问题。后续研究聚焦于构建新型高级语言及建立通用编程方法论的思想。然而我们先前的研究存在一个空白：未描述复杂度为多项式级的递归函数类。本工作为此类函数找到了充分条件。在诸多方面，本文的核心思想与我们证明甘迪不动点定理多项式模拟时采用的思路相似，但也存在显著差异。函数本身因其取值多样性而与谓词存在本质区别：谓词仅取真/假二值，而函数通常可取得多种值。即便取值数量有限，但若存在递归与简单乘法运算，计算过程中仍可能出现幂次与阶乘，这显然可能破坏函数的多项式计算复杂度。因此，在谓词扩展情形下证明甘迪不动点定理的多项式模拟后的三年间，我们始终面临这样的难题：如何为递归函数寻找既足够宽松以保持函数类的广泛性，又足够严格以确保不超越多项式界限的限制条件。