INVESTIGATION OF ALGEBRAIC GEOMETRY FOR COMPUTATIONAL APPLICATIONS

Mathematical branch known as algebraic geometry studies the geometric characteristics of polynomial problem solutions. It offers strong methods and tools for deciphering the structure of algebraic varieties, which are collections of answers to systems of polynomial equations. Algebraic geometry has attracted increasing attention in recent years for computational applications in a variety of fields. The goal of this study is to examine algebraic geometry's potential for use in computation. The main objective is to use the algebraic varieties' inherent geometric properties to create effective algorithms and approaches for handling challenging computing issues. New approaches for resolving difficult computational problems can be opened up by utilising the linkages between algebraic geometry and other branches of computer science, like optimisation, machine learning, cryptography, and robotics. The examination focuses on the fundamental ideas of algebraic geometry, such as Grobner bases, affine and projective varieties, algebraic curves and surfaces, and projective and affine varieties. These ideas form the foundation for creating computer algorithms based on algebraic geometry. To widen the range of applications, more sophisticated methods like computational invariant theory, numerical algebraic geometry, and tropical geometry are studied. By combining the robust theoretical foundation of algebraic geometry, the findings of this inquiry will boost computational methods. The outcomes are anticipated to be advantageous in a number of domains, including computer vision, data science, computer graphics, and encryption. Additionally, the inquiry might find brand-new links between computational issues and algebraic geometry, resulting in new understandings and advancements in both computer science and mathematics