Constructive Commutative Algebra: Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases (Lecture Notes in Mathematics)

Ihsen Yengui

  • 出版商: Springer
  • 出版日期: 2015-12-12
  • 售價: $2,330
  • 貴賓價: 9.5$2,214
  • 語言: 英文
  • 頁數: 284
  • 裝訂: Paperback
  • ISBN: 3319194933
  • ISBN-13: 9783319194936
  • 相關分類: UML
  • 海外代購書籍(需單獨結帳)

商品描述

The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring.

Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented.

Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.

商品描述(中文翻譯)

本書的主要目標是在交換代數中找到抽象證明中隱藏的建設性內容,特別是關於多項式環上專案模組(主要是Quillen-Suslin定理)和係數為評值環的多變量多項式的syzygies的著名定理。本書還提供了一些關於多項式環上專案模組理論的簡單且建設性的證明,並對Hermite環和Gröbner環猜想的最新進展提供了一些燈光。我們提出了一些關於單模完成的新猜想,這些猜想是由我們對單模完成問題的建設性方法引起的。

建設性代數可以被理解為計算代數的第一個預處理步驟,它可以引導出一般算法的發現,即使有時它們並不高效。從邏輯的角度來看,動態評估提供了抽象代數中兩個高度非建設性工具的建設性替代:排中律和Zorn引理。例如,為了在Dedekind環中構造一個理想的完全素因子分解,需要使用這些工具,而動態方法揭示了這個構造的計算內容。這些講義遵循這種動態哲學。