Constructive characterization of the reduction numbers

An elementary and constructive proof of Grothendieck's generic freeness lemma

We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully constructive, not using the axiom of choice or even the law of excluded middle. It was found by unwinding the result of a general topos-theoretic technique.

Constructive description of monogenic functions in a finite-dimensional commutative associative algebra

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,e_3$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the sense of Gateaux) functions of the variable $xe_1+ye_2+ze_3$ where $x,y,z$ are real, and obtain a constructive description of all mentioned functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux de…

$\mathbb{Z}[X]$ 中极大理想的物理解释与构造性分析

Material Interpretation and Constructive Analysis of Maximal Ideals in $\mathbb{Z}[X]$

本文提出物理解释方法，通过将经典蕴含转化为构造性析取，以 $\mathbb{Z}[X]$ 中极大理想为例给出构造性证明，揭示计算内容并探讨其在多个经典引理及算法实现中的应用。