Computational Arithmetic and the Geometry of Algebraic Curves

How do modern cryptographic systems and symbolic computation engines process complex geometric shapes? Understanding the intersection of computer science and algebraic geometry is essential for solving advanced computational problems. This course guides you from the fundamental definitions of fields and polynomials to the core algorithms used to analyze algebraic curves. You will build a strong mathematical foundation and learn how these theoretical concepts apply to modern computational challenges. What you'll learn: Understand the foundational algebraic structures and polynomial rings; Analyze the geometric properties of algebraic curves and their intersections; Apply computational arithmetic techniques to solve system equations; Learn modern algorithms for curve parametrization and singularity analysis; Explore how algebraic curves underpin modern cryptography and error-correcting codes; Practice symbolic computation methods through structured written exercises. We begin with essential terminology, defining fields, polynomials, and affine spaces. From there, you will progress through elimination theory, Bézout's theorem, and the algorithmic tools used to study curves. This text-based course is designed for computer science students, mathematicians, and software engineers looking for a clear, structured introduction to computational algebraic geometry. No advanced prerequisites are required. Start mastering the mathematics of algebraic curves today.