Some Applications of Gr¨ obner bases Hassan Noori, Abdolali Basiri, and Sajjad Rahmany Abstract—In this paper we will introduce a brief introduction to theory of Gr¨ obner bases and some applications of Gr¨ obner bases to graph coloring problem, automatic geometric theorem proving and cryptography. Keywords—Gr¨ obner bases, Application of Gr¨ obner bases, Auto- matic Geometric Theorem Proving, Graph Coloring, Cryptography. I. I NTRODUCTION We know from the Hilbert Basis Theorem that any ideal I in a polynomial ring over a field is finitely generated. However, through of all generators for I, we try to find the best generators to describe the ideal. More precisely we are looking for a generator G for I in order to answer the following questions [2], [8]: 1) Equality of ideals: Reduced Gr¨ obner bases are unique for any given ideal and monomial ordering, and also often computable in practice. Thus one can determine if two ideals I, J are equal by looking at their reduced Gr¨ obner bases. 2) Ideal membership problem: Let R = K[x 1 ,...,x n ] be a polynomial ring, given an ideal I ∈ R where I = 〈f 1 ,...,f s 〉, and given f ∈ R, determine whether f ∈ I? If so, can we compute h 1 ,...,h s ∈ R such that f = h 1 f 1 + ··· + h s f s ? To do this, we compute a Gr¨ obner basis G for I, then f ∈ I if and only if the reminder of the dividing f by G is 0. 3) Solving a system of polynomial equations: One of the most important applications of Gr¨ obner basis is the solving of a system of polynomial equations          f 1 (x 1 ,...,x n ) =0 f 2 (x 1 ,...,x n ) =0 . . . f m (x 1 ,...,x n ) =0 (1) To do this, at the first we compute a Gr¨ obner basis G = {g 1 ,g 2,1 ,g 2,2 ,...,g 2,r2 ,...,g n,1 ,g n,2 ,...,g n,rn } for the ideal generated by f 1 ,f 2 ,...,f m with respect to lexicographical order. In general we obtain the following form for our equations:          g 1 (x 1 )=0 g 2,1 (x 1 ,x 2 )=0,...,g 2,r2 (x 1 ,x 2 )=0 . . . g n,1 (x 1 ,...,x s )=0,...,g n,rn (x 1 ,...,x s )=0 (2) School of Mathematics and Computer Science, Damghan University nouri.hasan@gmail.com, {basiri,s rahmany}@du.ac.ir Research supported by the Research Affairs of Damghan University It is often easy to compute the solutions of the later system of polynomial equations. 4) Existence of solutions: The system of polynomial equations 1 has a solution if and only if the Gr¨ obner basis of {f 1 ,...,f m } is not equal to {1}. 5) Number of solutions: The system of polynomial equations 1 has a finite number of solutions if and only if any Gr¨ obner basis of {f 1 ,...,f m } has the following property: For every variable x i , there exists a polynomial such that its leading term with respect to the chosen term ordering is a power of x i . To compute a Gr¨ obner basis we need a division algorithm in K[x 1 ,...,x n ], like the division algorithm in K[x]. Unfortu- nately since there are multiple variables and multiple divisors, the remainder of this division is not unique. Hence if the remainder of the division of f by f 1 ,...,f m is equal to zero then f is in the ideal generated by f 1 ,...,f m , but if the remainder is not equal to zero we don’t know whether f is in the ideal generated by f 1 ,...,f m ? However, if we choose a good divisor, then the remainder is unique regardless of the order of the divisors. These divisors are called a Gr¨ obner basis. In order to define a Gr¨ obner basis, we first need to intro- duce some notations. If we fix a term order ≻, then every polynomial f has a unique leading monomial denoted by LM (f )= x α , this is the largest monomial x α with respect to the term order ≻ which occurs with nonzero coefficient in the expansion of f . The coefficient of the leading monomial x α is called the leading coefficient of f and denoted by LC(f ), finally the leading term of f is defined by LT (f )= LC(f )LM (f ) [8]. Definition 1 (Gr¨ obner basis) A Gr¨ obner basis for an ideal I in K[x 1 ,...,x n ] is a generating set G = {g 1 ,...,g m } such that the set {LT (g i ):1 ≤ i ≤ n} is a generator set for the ideal generated by LT (I )= {LT (f ): f ∈ I} [8]. If the monomial order ≻ is fixed, then every ideal I in K[x 1 ,...,x n ] has a unique reduced Gr¨ obner basis. There is some algorithms to compute the Gr¨ obner bases, B. Buchberger presented a such algorithm in his PhD thesis [7]. Later, Faugere presented the F 4 and F 5 algorithms, which are improved versions of the principal Buchberger algorithm [9], [10]. II. THE 3-COLORABLE PROBLEM There is a well-known problem in graph theory called the 3-color problem. Given a graph, we would like to know that if it can be three colored. Specifically, let G be a graph with n World Academy of Science, Engineering and Technology 77 2011 141