标题： 一个交换微分环的积分微分闭包 显示英文标题

标题： The integro-differential closure of a commutative differential ring

摘要： 一个积分微分环是一个在满足微积分基本定理的积分操作下闭合的微分环。 通过牛顿-莱布尼兹公式，广义求值是通过积分和微分定义的。 所诱导的求值不一定具有乘法性，这允许对具有奇点的函数进行建模，并导致广义的交错关系。 一般来说，并不是微分环中的每个元素都在同一环中具有原函数。 从一个交换微分环和可积与不可积元素的直接分解出发，我们构造了自由积分微分环。 这个积分微分闭包包含原始微分环中所有嵌套积分。 我们展示了嵌套积分乘积的广义求值所满足的关系。 研究这些常数的关系，我们通过利翁德词来描述某些乘积的求值，这些求值决定了其他所有求值。 我们还分析了自由积分微分环与交错代数之间的关系。 为了在原始微分环的积分微分闭包中的计算中保留积分，我们引入了拟积分微分环的概念，并给出了自由积分微分环的适应性构造。 最后，在给定的积分微分环中，我们考虑微分子环的内部积分微分闭包，并将其识别为自由积分微分环对某些常数的商。 显示英文摘要

摘要： An integro-differential ring is a differential ring that is closed under an integration operation satisfying the fundamental theorem of calculus. Via the Newton--Leibniz formula, a generalized evaluation is defined in terms of integration and differentiation. The induced evaluation is not necessarily multiplicative, which allows to model functions with singularities and leads to generalized shuffle relations. In general, not every element of a differential ring has an antiderivative in the same ring. Starting from a commutative differential ring and a direct decomposition into integrable and non-integrable elements, we construct the free integro-differential ring. This integro-differential closure contains all nested integrals over elements of the original differential ring. We exhibit the relations satisfied by generalized evaluations of products of nested integrals. Investigating these relations of constants, we characterize in terms of Lyndon words certain evaluations of products that determine all others. We also analyze the relation of the free integro-differential ring with the shuffle algebra. To preserve integrals in the original differential ring for computations in its integro-differential closure, we introduce the notion of quasi-integro-differential rings and give an adapted construction of the free integro-differential ring. Finally, in a given integro-differential ring, we consider the internal integro-differential closure of a differential subring and identify it as quotient of the free integro-differential ring by certain constants.