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Algorithmic Algebra

# E0 321: AlgorithmicAlgebra

#### Instructor: Ambedkar Dukkipati

Description | Announcements | Lectures | Assignments | Seminars

## Announcements

• 18 Aug 08 : I posted the first assignment.
• 06 Aug 08 : Class schedule:
Monday: 11.15 to 12.45 in Room No. 252
Wednesday: 3.30 to 5 in Room No. 252
Saturday: 11 to 12.30 in Room No. 252 (Tutorials and Extra Classes)

## Lectures

• 06 AUG 08: Introduction to Algorithmic Algebra, examples for solving system of Linear equations, division in k[x], 3-color problem as a decision problem in polynomial equations in several variables
Homework: Read Introduction chapter from, Algorithmic Algebra by Bubhaneshwar Mishra. Also try going through "Historical Ramblings in Algebraic Geometry and Related Algebra" by Shreeram S. Abhyankar, The American Mathematical Monthly, Vol. 83, No. 6 (Jun. - Jul., 1976), pp. 409-448. From here.

• 11 AUG 08: examples for division in k[x1,...xn], idea of Grobner bases , Equivalence relations.

• 13 AUG 08: Introduction to Groups, construction of Z_{n} (for this, I will use division algorithm for integers, but proof of this I will do it in some extra class before which I will cover Axioms for Natural Numbers, Mathematical Induction, Construction of Integers); Subgroups

• 20 AUG 08: Cosets, Group homomorphism, Isomorphism theorems, Def. of rings.

• 23 AUG 08 (Extra Class): Axioms for Natural Numbers, Mathematical Induction, Construction of Integers.

• 25 AUG 08: Introduction to Ring Theory.

• 1 SEP 08: Quotient Rings, Construction of Finite Fields

• 6 SEP 08: Introduction to Algebra-Geometry dictionary

• 15 SEP 08: Introduction to Algebra-Geometry dictionary (Contd...)

• 17 SEP 08: Principle Ideal domains and k[x]

• 19 SEP 08: Implicit and Parametric descriptions, Ideal membership problem and its applications.

• 22 SEP 08: Monomial Orders, Monomial Ideals

• 27 SEP 08: Monomial Ideal, Dickson Lemma (I will give the proof of Dickson Lemma, afterwords when we start doing abstract relations and orderings. You can go ahead read the proof from Cox at al)

• 29 SEP 08: Hilbert Bases Theorem (HBT) and Grobner Bases

• 1 OCT 08: Ascending Chain Condition (ACC) and HBT

• 6 OCT 08: Several characterizations of Grobner bases and Buchberger Criterion (I will prove this result later when we go back and started studying k[x1,...xn])

• 10 OCT 08: Buchberger Algorithm, Examples

• 13 OCT 08: Reduced Grobner Bases and corresponding Algorithms, Elimination term order, Elimination theorem, computing Grobner basis for intersection of two ideals

• 17 OCT 08: k-vector basis of "k[x1,...,xn]/ideal" i.e., residue class ring modulo ideal.

• 20 OCT 08: Monoid Rings

• 22 OCT 08: Monoid Rings (Cond..)

• 24 OCT 08: Monoid Rings: Universal property of Monoid Rings (Cond..)

• 12 Nov 08: Monoid Rings: Universal property of Polynomial Rings (Cond...)

• 14 Nov 08: Abstract relations and Towards Dicksons Lemma (Next few lectures we aim to prove much stronger version of Dicksons Lemma than what we have stated that "every monomial ideal is finitely generated")

• 17 Nov 08: Abstract relations and Towards Dicksons Lemma (Contd...)

• 19 Nov 08: Abstract relations Towards Dicksons Lemma (Contd...)

• 21 Nov 08: Finally we have Dicksons Lemma (Much much general form)

• 24 Nov 08: Modules, Noetherian Modules

• 26 Nov 08: Syzygies (Some hand waiving on applications of Grobner bases to Integer Programming...I am sorry for this...but I will give printed material to read on your own)

• 28 Nov 08: Syzygies and Grobner bases...Proof of Buchberger Criterian. Universal Grobner bases.

## Seminars

• Sudan's List decoding of Reed-Soloman's Codes: A Grobner bases Perspective by D. Harish
• Grobner Bases and Graph Colorings by Manu B
• Groebner bases for modules and their application in Cryptography by C. Srikant
• Grobner basis view of Welch-Berlekamp algorithm by Tejas S. P
• Genetic Regulatory Networks by Vikas Kumar Garg
• Grobner Bases in Algebraic Statistics by Abyay Kumar Yadav