Constructive Commutative Algebra: Projective Modules Over Polynomial Rings And Dynamical Gröbner Bases (lecture Notes In Mathematics)
by Ihsen Yengui /
2015 / English / PDF
2.7 MB Download
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.
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.
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.
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.