Plan for Lectures
Topics to be Covered
- Basic Algebraic Primitives (2 lectures)
- Polynomial Multiplication, Evaluation, Interpolation (1 lecture).
- Discrete Fourier Transform (1 lecture).
- Univariate Polynomial Division (1 lecture).
- Chinese Remainder Algorithm (1 lecture).
- Resultants, and Modular GCD Algorihtms (1 lecture)
- Factoring Univariate Polynomials over Finite Fields (2 lectures).
- Factoring Univariate Polynomials over the Rationals (2 lectures).
- Factoring Bivariate Polynomials (1 Lecture)
- Application of factoring: list-decoding of Reed-Solomon codes (1 lecture)
- Introduction to Commutative Algebra & Algebraic Geometry (1 lecture)
- Groebner Bases, Multivariate Polynomial Division Algorithm, Buchberger’s Algorithm (2 lectures)
- Computational Invariant Theory (2 lectures)
- Fast Linear Algebra (4 lectures)
- Selected Topics and Conclusion
Lecture Schedule
| Date | Topics | Slides | |
|---|---|---|---|
| Lecture 0 | January 11 | Introduction & Overview of Course | |
| Lecture 1 | January 11 | Basic Polynomial Arithmetic and Basic Algebraic Operations | PDF and PDF |
| Lecture 2 | January 13 | Algebraic Models of Computation | |
| Lecture 3 | January 18 | Evaluation, Interpolation and Multiplication of Polynomials | |
| Lecture 4 | January 20 | Discrete Fourier Transform | |
| Lecture 5 | January 25 | Univariate Polynomial Division | |
| Lecture 6 | January 27 | Chinese Remaindering Theorem | |
| Lecture 7 | February 1 | Resultants & Modular GCD Algorithm | |
| Lecture 8 | February 3 | Univariate Polynomial Factoring over Finite Fields I | |
| Lecture 9 | February 8 | Univariate Polynomial Factoring over Finite Fields II | |
| Lecture 10 | February 10 | Univariate Polynomial Factoring over the Integers | |
| Lecture 11 | February 22 | Small Vectors in a Lattice - Lenstra, Lenstra, Lovasz Algorithm | |
| Lecture 12 | February 24 | Introduction to Commutative Algebra & Algebraic Geometry | |
| Lecture 13 | March 1 | Multivariate Polynomial Division Algorithm & Monomial Ideals | |
| Lecture 14 | March 3 | Groebner Bases & Buchberger’s Algorithm | |
| Lecture 15 | March 8 | Introduction to Invariant Theory | |
| Lecture 16 | March 10 | Reynolds Operator & Finite Generation of Invariant Rings | |
| Lecture 17 | March 17 | Bivariate Polynomial Factoring | |
| Lecture 18 | March 22 | Application: List-Decoding of Reed Solomon Codes | |
| Lecture 19 | March 24 | Matrix Multiplication & Modular Composition | |
| Lecture 20 | March 29 | Partial Derivatives & Exponent of Linear Algebra | |
| Lecture 21 | March 31 | Linearly Recurrent Sequences | |
| Lecture 22 | April 5 | Black-Box Linear Algebra & Wiedemann’s Algorithm for Linear System Solving | |
| Lecture 23 | April 7 | Elimination Ideals and Implicitization | |
| Lecture 24 | April 12 | Complexity of Ideal Membership Problem | |
| Lecture 25 | April 14 | Conclusion |
Suggested Reading
| Topics | Suggested Reading | |
|---|---|---|
| Lecture 0 | Introduction & Overview of Course | notes |
| Lecture 1 | Basic Polynomial Arithmetic and Basic Algebraic Operations | Arne’s notes |
| Lecture 2 | Algebraic Models of Computation | SY10 chapters 1 & 2 |
| Lecture 3 | Evaluation, Interpolation and Multiplication of Polynomials | Arne’s notes |
| Lecture 4 | Discrete Fourier Transform | Arne’s notes |
| Lecture 5 | Univariate Polynomial Division | Arne’s notes |
| Lecture 6 | Chinese Remaindering Theorem and Algorithm | Arne’s notes |
| Lecture 7 | Resultants & Modular GCD Algorithm | Arne’s notes |
| Lecture 8 | Univariate Polynomial Factoring over Finite Fields I | Madhu’s notes - lecture 5 |
| Lecture 9 | Univariate Polynomial Factoring over Finite Fields II | Madhu’s notes - lecture 6 |
| Lecture 10 | Univariate Polynomial Factoring over the Rationals I | Madhu’s notes - lecture 10 |
| Lecture 11 | Univariate Polynomial Factoring over the Rationals II | Madhu’s notes - lecture 11 |
| Lecture 12 | Introduction to Commutative Algebra & Algebraic Geometry | CLO'15 - chapters 1 & 4 |
| Lecture 13 | Multivariate Polynomial Division Algorithm & Monomial Ideals | CLO'15 - chapter 2 |
| Lecture 14 | Groebner Bases & Buchberger’s Algorithm | CLO'15 - chapter 2 |
| Lecture 15 | Introduction to Invariant Theory | S'08 - chapters 1 & 2 |
| Lecture 16 | Reynolds Operator & Finite Generation of Invariant Rings | S'08 - chapter 2 |
| Lecture 17 | Bivariate Polynomial Factoring | Madhu’s notes - lectures 7-9 |
| Lecture 18 | Application: List-Decoding of Reed Solomon Codes | Venkat’s survey: Algorithmic results in list decoding |
| Lecture 19 | Matrix Multiplication & Modular Composition | [vzGG] - Chapter 12 |
| Lecture 20 | Partial Derivatives & Exponent of Linear Algebra | Schost’s notes |
| Lecture 21 | Linearly Recurrent Sequences | [vzGG] - Chapter 12 and Schost’s notes |
| Lecture 22 | Wiedemann’s Algorithm for Linear System Solving Over Finite Fields | [vzGG] - Chapter 12 and Schost’s notes |
| Lecture 23 | Elimination Ideals and Implicitization | CLO'15 - chapter 3 |
| Lecture 24 | Complexity of Ideal Membership Problem | Madhu’s notes - lecture 14 |
| Lecture 25 | Conclusion | no notes |