COURSE UNIT TITLE

: COMPUTATIONAL COMMUTATIVE ALGEBRA I

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4065 COMPUTATIONAL COMMUTATIVE ALGEBRA I ELECTIVE 4 0 0 7

Offered By

Mathematics (English)

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (Evening)
Mathematics (English)

Course Objective

The aim of this course is to introduce methods of commutative algebra with a view towards algebraic geometry and with the computational methods using Groebner basis.

Learning Outcomes of the Course Unit

1   Will be able to find Groebner basis of a given ideal in the polynomial ring over a field with n indeterminates.
2   Will be able to determine whether a given polynomial is in a given ideal in the polynomial ring over a field with n indeterminates using Groebner basis and division algorithm.
3   Will be able to solve a given system of polynomial equations over a field with n indeterminates under suitable conditions using the Elimination Theorem and the Extension Theorem.
4   Will be able to find the smallest variety containing a set defined by polynomial parametrizations by using the Elimination Theorem.
5   Will be able to interpret the relation between ideals and varieties using the ideal-variety correspondence and the Hilbert s Nullstellensatz when the field is algebraically closed.
6   Will be able to interpret the coordinate ring of an affine variety as a quotient ring of the polynomial ring.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Affine varieties. Parametrizations of affine varieties. Ideals.
2 Monomial orders. Division algorithm. Dickson's Lemma for monomial ideals. The Hilbert Basis Theorem.
3 Groebner bases. Buchberger's Algorithm.
4 Applications of Groebner bases.
5 Elimination and Extension Theorems.
6 The geometry of eliminination. Implicitization.
7 Singular points and envelopes.
8 Resultants.
9 The Extension Theorem.
10 Hilbert's Nullstellensatz. Radical ideals and the Ideal-Variety Correspondence, the algebra-geometry dictionary.
11 Sums, product and intersections of ideals. Zariski closure and quotients of ideals.
12 Irreducible varieties and prime ideals. Decomposition of a variety into irreducible varieties. Primary decomposition of ideals.
13 Polynomial mappings. Quotients of polynomial rings.
14 The coordinate ring of an affine variety. Rational functions on a variety. Closure Theorem.

Recomended or Required Reading

Textbook(s): Cox, D., Little, J. and OShea D. Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, Springer, 2007.
Supplementary Book(s): 1) Reid, M. Undergraduate Algebraic Geometry, Cambridge University Press, 1998.
2) Reid, M. Undergraduate Commutative Algebra, Cambridge University Press, 1995.
3) Greuel, G.-M. and Pfister, G. A Singular Introduction to Commutative Algebra, With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, Second, Extended Edition, Springer, 2008.
References: 1) Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry, Springer, 1995.
2) Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra, Addison Wesley, 1994.
3) Sharp, R. Y. Steps in Commutative Algebra, Second edition, Cambridge University Press, 2004.
4) Matsumura, H. Commutative Ring Theory, Cambridge University Press, 1989.

Planned Learning Activities and Teaching Methods

Lecture notes, presentation, problem solving, discussion.

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 VZ Vize
2 FN Final
3 BNS BNS VZ * 0.40 + FN * 0.60
4 BUT Bütünleme Notu
5 BBN Bütünleme Sonu Başarı Notu VZ * 0.40 + BUT * 0.60


Further Notes About Assessment Methods

1 Midterm Exam
Final Exam

Assessment Criteria

%40 (Midterm examination) +%60 (Final examination)

Language of Instruction

English

Course Policies and Rules

You can be successful in this course by studying from your textbooks and lecture notes on the topics to be covered every week, coming to class by solving the given problems, establishing the concepts by discussing the parts you do not understand with your questions, learning the methods, and actively participating in the course.

Contact Details for the Lecturer(s)

Engin Mermut
e-mail: engin.mermut@deu.edu.tr
Phone: (232) 301 85 82

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 4 56
Preparations before/after weekly lectures 14 5 70
Preparation for midterm exam 1 20 20
Preparation for final exam 1 25 25
Final 1 2 2
Midterm 1 2 2
TOTAL WORKLOAD (hours) 175

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.1535333
LO.2535333
LO.3535333
LO.4535333
LO.5535333
LO.6535333

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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