Analyzing the Asymptotic Properties of Zero Sets of Multivariate Polynomials and Their Practical Applications

Abstract

The study of the asymptotic properties of zero sets of multivariate polynomials holds significant importance in various mathematical disciplines and finds practical applications in diverse fields. The zero sets, also known as algebraic varieties, are fundamental objects in algebraic geometry, providing insights into the geometric structure and solutions of polynomial equations in multiple variables. This paper explores the theoretical analysis of the asymptotic behavior of zero sets as the degrees of the polynomials increase, revealing crucial insights into the geometry of these sets in high-dimensional spaces. We investigate how the number of isolated zeros and their distribution change with increasing polynomial degrees, shedding light on the limiting behavior of algebraic varieties. the practical applications of these findings extend to numerous fields, including robotics, computer-aided design, signal processing, and cryptography. The knowledge of the asymptotic properties of zero sets enables efficient algorithm design and optimization in solving polynomial systems, leading to enhanced performance and accuracy in various computational tasks.