Computational Commutative Algebra and Algebraic Geometry: Course and exercises with detailed solutions

This book is intended to provide material for a graduate course of one or two semesters on computational commutative algebra and algebraic geometry spotlighting potential applications in cryptography. Also, the topics in this book could form the basis of a graduate course that acts as a segue between an introductory algebra course and the more technical topics of commutative algebra and algebraic geometry. It contains a total of 124 exercises (15 on Gröbner bases over arithmetical rings, 11 on Varieties, Ideals and Gröbner bases, 19 on Finite fields, 11 on Algorithms for cryptography, 33 on Algebraic plane curves, and finally 35 on Elliptic curves) with detailed solutions as well as an important number of examples that illustrate definitions, theorems, and methods. This is very important for students or researchers who are not familiar with the topics discussed. Experience has shown that beginners who want to take their first steps in algebraic geometry are usually discouraged by the difficulty of the proposed exercises and the absence of detailed answers. So, exercises (and their solutions) as well as examples occupy a prominent place in this course. This book is not designed as a comprehensive reference work, but rather as a selective textbook. The many exercises with detailed answers make it suitable for use both in a math or computer science course.