Constructive Commutative Algebra: Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases (Lecture Notes in Mathematics)
暫譯: 建設性交換代數:多項式環上的射影模與動態格羅布納基礎(數學講義)
Ihsen Yengui
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)
這本書的主要目標是尋找隱藏在具體定理的抽象證明中的建設性內容,特別是與多項式環上的專項模(主要是 Quillen-Suslin 定理)和具有值域環係數的多變數多項式的 syzygies 相關的著名定理。
本書還提供了一些多項式環上專項模理論結果的簡單且建設性的證明,並對 Hermite 環和 Gröbner 環猜想的最新進展進行了闡述。基於我們對單模完成問題的建設性方法,提出了新的單模完成的猜想。
建設性代數可以被理解為計算代數的第一步預處理,這導致了通用算法的發現,即使這些算法有時並不高效。從邏輯的角度來看,動態評估為抽象代數的兩個高度非建設性的工具提供了一個建設性的替代方案:排中律和佐恩引理。例如,這些工具在構造 Dedekind 環中理想的完整質因數分解時是必需的,而動態方法則揭示了這一構造的計算內容。這些講義遵循這一動態哲學。