In the modern landscape of mathematics, data science, cryptography, and quan- tum information, algebraic methods play a foundational role. Yet many of the structures we study—polynomial rings, number fields, ideals, and varieties—are too complex to handle without computational assistance. Computational algebra bridges the abstract world of algebra with the practical power of algorithmic computation. This course is designed to introduce both the theory and practice of symbolic computation, with an eye toward applications in quantum computing and emerging technologies.
Contents
Chapter 1 — Introduction to Computational Algebra  - Symbolic computation  - Computer Algebra Systems (CAS)  - Complexity basics
Chapter 2 — Polynomial Arithmetic and Data Structures  - Dense and sparse representations  - Basic operations (Karatsuba, FFT)  - Evaluation and interpolation  - Python project (SymPy/Sage)
Chapter 3 — Greatest Common Divisors and Euclidean Algorithms  - Euclidean algorithm (integers & polynomials)  - Subresultants, half-GCD  - Python project
Chapter 4 — Projective Spaces and Weighted Projective Spaces  - Projective spaces  - Normalization of homogeneous coordinates  - Weighted projective spaces  - Weighted gcd and normalization
Chapter 5 — Gröbner Bases and Buchberger’s Algorithm  - Ideals and monomial orders  - Buchberger’s algorithm  - Elimination theory  - Gröbner bases for systems  - Python project
Chapter 6 — Introduction to Quantum Computing  - Qubits, quantum gates, circuits  - Linear algebra for quantum computing
Chapter 7 — Quantum Algorithms Basics  - Quantum Fourier Transform (QFT)  - Grover’s algorithm  - Shor’s algorithm
Chapter 8 — Hidden Subgroup Problem and Group-Theoretic Algorithms  - HSP framework  - Applications in number theory  - Group-theoretic extensions
Chapter 9 — Quantum Linear Algebra  - HHL algorithm  - Quantum singular value decomposition (QSVD)  - Applications to algebra
Chapter 10 — Lattice Reduction and Quantum Attacks  - Lattice basics  - LLL algorithm  - Quantum approaches to lattice problems
Chapter 11 — Advanced Quantum Algebraic Algorithms  - Quantum walks on graphs  - Nonlinear polynomial systems  - Current research directions
Chapter 12 — Applications and Quantum Algebraic Geometry  - Quantum error correction and codes  - Cryptography and security implications
Chapter 2 — Polynomial Arithmetic and Data Structures  - Dense and sparse representations  - Basic operations (Karatsuba, FFT)  - Evaluation and interpolation  - Python project (SymPy/Sage)
Chapter 3 — Greatest Common Divisors and Euclidean Algorithms  - Euclidean algorithm (integers & polynomials)  - Subresultants, half-GCD  - Python project
Chapter 4 — Projective Spaces and Weighted Projective Spaces  - Projective spaces  - Normalization of homogeneous coordinates  - Weighted projective spaces  - Weighted gcd and normalization
Chapter 5 — Gröbner Bases and Buchberger’s Algorithm  - Ideals and monomial orders  - Buchberger’s algorithm  - Elimination theory  - Gröbner bases for systems  - Python project
Chapter 6 — Introduction to Quantum Computing  - Qubits, quantum gates, circuits  - Linear algebra for quantum computing
Chapter 7 — Quantum Algorithms Basics  - Quantum Fourier Transform (QFT)  - Grover’s algorithm  - Shor’s algorithm
Chapter 8 — Hidden Subgroup Problem and Group-Theoretic Algorithms  - HSP framework  - Applications in number theory  - Group-theoretic extensions
Chapter 9 — Quantum Linear Algebra  - HHL algorithm  - Quantum singular value decomposition (QSVD)  - Applications to algebra
Chapter 10 — Lattice Reduction and Quantum Attacks  - Lattice basics  - LLL algorithm  - Quantum approaches to lattice problems
Chapter 11 — Advanced Quantum Algebraic Algorithms  - Quantum walks on graphs  - Nonlinear polynomial systems  - Current research directions
Chapter 12 — Applications and Quantum Algebraic Geometry  - Quantum error correction and codes  - Cryptography and security implications
This course is run entirely on Python using SymPy and Sagemath