International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-9 Issue-1, October 2019 1092 Published By: Blue Eyes Intelligence Engineering & Sciences Publication Retrieval Number: A9486109119/2019©BEIESP DOI: 10.35940/ijeat.A9486.109119 Abstract: This paper is a survey on Groebner basis and its applications on some areas of Science and Technology. Here we have presented some of the applications of concepts and techniques from Groebner basis to broader area of science and technology such as applications in steady state detection of chemical reaction network (CRN) by determining kinematics equations in the investigation and design of robots. Groebner basis applications could be found in vast area in circuits and systems. In pure mathematics, we can encounter many problems using Groebner basis to determine that a polynomial is invertible about an ideal, to determine radical membership, zero divisors, hence so forth. A short note is being presented on Groebner basis and its applications. Keywords: Groebner basis, polynomials, polynomials rings, ideals, division algorithm, applications of Groebner basis. I. INTRODUCTION Polynomial systems are available in many areas of mathematics, natural sciences and engineering, and the theory of Groebner basis is one of the most advance tools for solving polynomial systems. Groebner basis are the basis for the finitely generated ideals of a polynomial ring with several variables. The three steps monomial ordering, division algorithm and Hilbert basis theorem are the shortest way which leads to the definition of Groebner basis [15]. The notion of Groebner basis was introduced by Buchberger in 1965 to describe ideals of commutative algebras. This notion generalizes both the Gaussian elimination method for solving systems of polynomial equations and the division of polynomials in multiple indeterminates. In the case of systems of linear polynomial equations, Groebner basis give the Gaussian elimination algorithm. In the case of the division, of a polynomial by a set of polynomials, the remainder is not always unique. Hence if the remainder of the division of by is equal to zero then is in the ideal generated by , but if the remainder is not equal to zero we cannot determine whether is in the ideal generated by . Also, if we choose any divisor and the remainder is unique regardless of the order of divisors. These divisors are the Groebner basis. Groebner basis can also be used to decide whether or not an element of a commutative algebra is in some ideal. Given an ideal of the ring of polynomials in indeterminates, a Groebner basis of gives us an algorithm deciding whether or not an element of is in A Groebner basis is a set of multivariate Revised Manuscript Received on October 15, 2019 * Correspondence Author Yengkhom Satyendra Singh, Department of Mathematics, School of Applied Sciences, Reva University, Bangalore, India. Email: yengkhom123@gmail.com Benaki Lairenjam, Department of Mathematics, School of Applied Sciences Reva University, Bangalore, India. Email: benaki_lai@yahoo.co.in nonlinear polynomials that reduces many fundamental problems in mathematics into simple algorithmic solutions such as finding intersection of ideal, finding Hilbert dimension and envelopes in commutative algebra; geometric theorem proving, graph coloring, summation and integration, linear integer optimization in the field of mathematics; also it has many applications in coding theory, robotics, software engineering and in petroleum industry for natural sciences and engineering [3]. Groebner basis theory in [1] and [2], is of great importance in various scientific fields with the progress in Computer Algebra System (CAS) such as mathematical modeling and simulations for problems. To name a few problems that the Groebner basis technique solved recently are determining genetic connection between the species, reverse engineering problems, inverse kinematics in robotics and to control oil platforms using artificial intelligence. II. PRELIMINARIES Let indicate a field generally C Q, R, or p Z where p is prime. Let represent the polynomial ring in the indeterminates over the field . A monomial is nothing but a product of the variables, that is where . Furthermore, the total degree of the monomial could be define as . A polynomial of can be define as a linear combination of monomials such as . That is all the polynomials of is of the form where is the degree of . Groebner Basis and its Applications Yengkhom Satyendra Singh, Benaki Lairenjam