Advancements in Computational Algebraic Geometry: Techniques and Applications

Computational algebraic geometry has experienced significant advancements in recent years, driven by both theoretical breakthroughs and practical applications comprehensive review of the latest techniques in computational algebraic geometry, highlighting their development and impact on various domains. We begin by discussing foundational methods, including Gröbner bases, resultants, and elimination theory, which have been pivotal in solving polynomial systems and understanding algebraic varieties emerging algorithms that leverage improvements in computational efficiency, such as homotropy continuation methods and numerical algebraic geometry techniques. the applications of these advancements across different fields. In robotics and computer vision, for example, algebraic geometry methods are applied to solve problems related to motion planning and object recognition. In cryptography, recent developments in computational algebraic geometry offer new approaches to designing secure cryptographic systems resistant to quantum attacks. We also discuss the integration of algebraic geometry with machine learning and data science, emphasizing its role in improving predictive modeling and pattern recognition.