On Groebner Bases and Their Use in Solving Some Practical Problems

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1 Universal Journal of Computational Mathematics 1(1): 5-14, 2013 DOI: /ujcmj On Groebner Bases and Their Use in Solving Some Practical Problems Matej Mencinger 1,2 1 University of Maribor, Faculty of Civil Engineering, Smetanova 17, 2000, Maribor, Slovenia 2 Institute of Mathematics, Physics and Mechanics, Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia *Corresponding Author: matej.mencinger@um.si Copyright 2013 Horizon Research Publishing All rights reserved. Abstract Groebner basis are an important theoretical building block of modern (polynomial) ring theory. The origin of Groebner basis theory goes back to solving some theoretical problems concerning the ideals in polynomial rings, as well as solving polynomial systems of equations. In this article four practical applications of Groebner basis theory are considered; we use Groebner basis to solve the systems of nonlinear polynomial equations, to solve an integer programming problem, to solve the problem of chromatic number of a graph, and finally we consider an original example from the theory of systems of ordinary (polynomial) differential equations. For practical computations we use systems»mathematica«and»singular«. Keywords Polynomial system of (differential) equations, integer linear programming, chromatic number of a graph, polynomial rings, Groebner basis, CAS systems 1. Introduction In his 1965 thesis, Bruno Buchberger[3] developed the theory of what we today call Groebner basis. The theory allows computations in multivariate polynomial rings analogous to those we use in single variable polynomial rings. The theory of Groebner basis can also be seen as a generalization of Gaussian elimination of a linear (polynomial) system, which yields the well-known row echelon form. However, applications of Groebner basis can be found in different fields of (mathematical) science. Roughly speaking, they can be used anywhere where some polynomial(s) (ideals) appear. The basic idea here was to generalize a step in the classical Gaussian elimination algorithm, when for example the pair of polynomials (obtained from equations) ff = 3xx + 7yy 5zz 2 and gg = 2xx + 3yy 8zz 6 is replaced by an (equivalent) pair of polynomials (equations) ff = 3xx + 7yy 5zz 2 and SS ff,gg = 5yy + 14zz Recall that the least common multiple of 3 and 2 is 6 and that SS ff,gg = 6 3 (3xx + 7yy 5zz 2) 6 (2xx + 3yy 8zz 6) 2 = 5yy + 14zz In the sequel we provide some definitions from the ring theory which help to understand the origin of Groebner basis (see e.g. [5] or [13] for details). To this end just recall, that the set of polynomials ff 1, ff 2,, ff ss implies a system of equations ff 1 (xx 1, xx 2,, xx nn ) = 0, ff 2 (xx 1, xx 2,, xx nn ) = 0,, ff ss (xx 1, xx 2,, xx nn ) = 0 on one hand, and is naturally associated with the ideal ss (1.1) ff 1, ff 2,, ff ss = h jj ff jj : h 1,, h ss kk[xx 1, xx 2,, xx nn ] jj =1 generated by polynomials ff 1, ff 2,, ff ss (which is also the basis of the ideal I = ff 1, ff 2,, ff ss ). The most common monomial term ordering is lexicographic; though many other are well-known, too (e.g. (graded) reverse lexicographic order, elimination order, etc.). Recall that as soon as the monomial order is chosen we can speak of leading monomial (LM), leading term(lt) and leading coefficient (LC) of the polynomial. Recall also, that any vector cc R nn defines a weight (monomial term) ordering < cc in R[xx 1, xx 2,, xx nn ] in the following way: xx αα < cc xx ββ cc αα < cc ββ oooo cc αα = cc ββ aaaaaa αα < llllll ββ, where cc αα stands for the standard dot product. For example, if cc = (1,5,10), we have xx 5 1 xx 1 2 xx 2 3 < cc xx 1 1 xx xx 3 since cc αα = (1,5,10) (5,1,2) = 30 and cc ββ = (1,5,10) (1,0,3) = 31. Note, that for < cc the leading term of gg = 2xx 5 1 xx 1 2 xx 2 3 5xx 1 1 xx xx 3 is LLLL(gg) = 5xx xx 3 while for < llllll the leading term of (the same) gg is LLLL(gg) = 2xx 5 1 xx 1 2 xx 2 3.

2 6 On Groebner Bases and Their Use in Solving Some Practical Problems Once a ring and a monomial order are chosen, one can divide polynomial by another (set of ) polynomial(s). The generalization of the Gaussian elimination process for solving system (1.1) requires to»divide a polynomial by a set of polynomials«. The well-known elementary row operations (from Gaussian elimination) are defined by the fact that on every step ss (of the Gaussian elimination process for ff 0 (xx ) = 0 ) the solution of changed system ff ss (xx ) = 0 remains the same. Let us recall the first example in context of notationff ss (xx ) = 0. The initial system:ff 0 = 3xx + 7yy 5zz 2,gg 0 = 2xx + 3yy 8zz 6 is replaced with ff 1 = ff 0, and gg 1 = SS ff,gg = 5yy + 14zz Note that gg 1 = 2ff 0 3gg Note also, that if ff 0 (xx ) = 0, gg 0 (xx ) = 0 for some xx, then gg 1 (xx ) = 0, too. The main idea is that one can replace the initial pair ff 0, gg 0 with ff 1, gg 1 if ff 1 and gg 1 can be»divided between«the initial polynomials, ff 0, gg 0 : ff 1 = 1ff 0 + 0gg 0 and gg 1 = 2ff 0 3gg 0 The division of a polynomial between a set of (other) polynomials is called the multivariable division and lead toward the definition of Groebner bases. 2. The Multivariable Division Algorithm Considering ff 1 = XXXX XX 3, ff 2 = XX + YY 2 and ff = XX 2 + XXXX + 2XX 3 and choosing the lexicographic order XX > YY, then we can easily verify that ff = 2ff 1 + (XX + 3YY YY 2 )ff 2 + ( 3YY 3 + YY 4 ) and on the other hand, when the»importance«of ff 1, ff 2 is chaned to 1. ff 2, 2. ff 1, we have: ff = (XX + 2XX 2 + YY YY 2 2XXYY 2 + 2YY 4 )ff 2 + 0ff 1 + ( 3YY 3 + YY 4 2YY 6 ). Obviously, this multivariable division is very sensitive on the order of ff 1, ff 2. The order affects the multi-quotients qq 1, qq 2, as well as the remainder rr. When dividing the polynomial ff with the (ordered) set FF = (ff 1, ff 2 ), one can write: ff = {qq 1, qq 2 }, rr instead of ff = qq 1 ff 1 + qq 2 ff 2 + rr. Using this notation, in the first case we have ff = { 2, XX + 3YY YY 2 }, 3YY 3 + YY 4 and in the second case we have ff = {XX + 2XX 2 + YY YY 2 2XXYY 2 + 2YY4,0, 3YY3+YY4 2YY6. Readily, if one chooses the lexicographic term order YY > XX the results would be different again, as one can observe from Fig. 1, where system MATHEMATICA is used for the multivariable division. Obviously, YY > XX gives»simpler«results than XX > YY (concerning the quotients) Figure 1. The MATHEMATICA results for multivariable division for different monomial orders. Concerning the problem of multivariable division we have the following result Theorem 2.1. Fix a monomial order > and letff = (ff 1, ff 2,, ff ss ). Then every polynomialff kk[xx 1, xx 2,, xx nn ] can be written as ff = qq 1 ff 1 + qq 2 ff qq ss ff ss + rr, where qq ii, rr kk[xx 1, xx 2,, xx nn ] and either rr = 0 or rr is a kk[xx 1, xx 2,, xx nn ] linear combination of monomials none of which are divisible by the leading terms of any offf 1, ff 2,, ff ss, which means that rr is reduced with respect to FF = {ff 1, ff 2,, ff ss } (i.e. rr has lower degree than any ofthe divisorsff 1, ff 2,, ff ss ). We can alternativelly write: ff FF The proof of the above theorem is based on the multivariable division algorithm, which can nowadays be found in any textbook of commutative algebra.it is sketched in Fig. 2 (see e.g. [13]). rr.

3 Universal Journal of Computational Mathematics 1(1): 5-14, Recall that the solutions of (1.1) is actually an affine variety defined by the ideal II = ff 1, ff 2,, ff ss. We would like to use the division algorithm for the question of ideal membership. If dividing ff by ff 1, ff 2,, ff ss gives a remainder of zero then we know ff II. But the converse is not true. Even if ff has a nonzero remainder there may be some ways to divide it in a different order that gives a remainder of zero, as we will see in the following example. (Note that the example from Fig.1 shows that the remainders are not unique.) Let us consider for example ff 1 = xx 2 1, ff 2 = xxxx + 2 and ff = xx 2 yy + xxxx + 2xx + 2 and choose the lexicographic order xx > yy, then we obtain ff = yyff 1 + ff 2 + (2xx + yy). The remainder rr = 2xx + yy 0, thus one could conclude that ff ff 1, ff 2, but if the order of divisors is changed to FF = (ff 2, ff 1 ), we have FF ff 0 ; namely ff = 0ff 1 + (xx + 1)ff = (xx + 1)ff and ff ff 1, ff 2 after all. Finally, let us consider ff 1 = xx + yy, ff 2 = xx yy and ff = 2yy in the ring R[xx, yy] and fix the lexicographic term order xx > yy. Then obviously ff = ff 1 ff 2 ff 1, ff 2, but since LLLL(xx + yy) = LLLL(xx yy) = xx, and because xx > yy the division algorithm from Fig. 2 returns the remainder rr = 2yy. As we shall see, Groebner bases are the solution to the above problems. Figure 2. Multivariable Division Algorithm. 3. Groebner Bases The Groebner basis is a special generating set for our ideals ff 1, ff 2,, ff nn for which the multivariable division algorithm for a given ff returns the remainder rr = 0 if and only if ff gg 1, gg 2,, gg tt. More precisely, the Groebner basis of an ideal II kk[xx 1, xx 2,, xx nn ] is a finite subset GG = {gg 1, gg 2,, gg tt } of II such that LLLL(II) = LLLL(gg 1 ), LLLL(gg 2 ),, LLLL(gg tt ). Every nonzero ideal II kk[xx 1, xx 2,, xx nn ] has the Groebner basis. Note that LLLL(II) = LLLL(gg): gg II {0} = LLLL(gg): gg II {0} is a monomial ideal and by Dickson's lemma (see [5]) LLLL(II) = LLLL(gg 1 ),LLLL(gg 2 ),,LLLL(gg tt ) = LT(gg 1 ),LT(gg 2 ),,LT(gg tt ) for some finite set gg ii II. Furthermore, due to the multivariable division algorithm, if ff II we have ff = qq 1 gg 1 + qq 2 gg qq ss gg ss + rr and no term of rr is divisible by any of LLLL(gg 1 ), LLLL(gg 2 ),, LLLL(gg tt ). Thus, rr = ff (qq 1 gg 1 + qq 2 gg qq ss. so LLLL(rr) LLLL(II) = LLLL(gg 1 ), LLLL(gg 2 ),, LLLL(gg tt ). But no term of rr is divisible by any of the LLLL(gg ii ) and so we must have rr = 0, which implies: ff LLLL(gg 1 ), LLLL(gg 2 ),, LLLL(gg tt ). The opposite implication is obvious. Thus, the Groebner basis is a basis.

4 8 On Groebner Bases and Their Use in Solving Some Practical Problems Obviously, if GG = {gg 1, gg 2,, gg tt } is the Groebner basis of II, the remainder of any ff II (after applying the multidivision algorithm) is unique. If ff = qq 1 gg 1 + qq 2 gg qq ss gg ss + rr and ff = qq 1 gg 1 + qq 2 gg qq ss gg ss + rr then rr rr = (qq 1 qq 1 )gg (qq ss qq ss )gg ss II. If rr rr 0 then LLLL(rr rr ) LLLL(II) which implies that LLLL(gg ii ) divides LLLL(rr rr ) for some ii. But this leads to a contradiction, since no term of rr or rr is divisible by any LLLL(gg ii ). Thus, we must have rr = rr and therefore gg = gg. Testing whether a basis is a Groebner basis is intimately connected with the so called SS polynomial for a given pair of polynomials ff, gg kk[xx 1, xx 2,, xx nn ]; a generalization of SS ff,gg polynomial defined in the introduction. The SS polynomial of ff and gg is defined as follows. Let ff, gg kk[xx 1, xx 2,, xx nn ] be nonzero polynomials. Find the least common multiple of their leading monomials:xx γγ = LLLLLL(LLLL(ff), LLLL(gg)). Then the SS polynomial of ff and gg is defined by: xxγγ xxγγ SS(ff, gg) = LLLL(ff) ff LLLL(gg) gg. Note, that the SS polynomials provide cancellacion of leading terms and in fact are the only way that cancelation happens among sums of terms of the same multi-degree. The Buchcberger's basic observation was the following criterion. Let II be an ideal. Then GG = {gg 1, gg 2,, gg tt } is a Groebner bases (for II ) if and only if for all ii jj the remainder on division of SS(gg ii, gg jj ) by GG is zero: SS(gg ii, gg jj ) GG 0 ii jj. This criterion is the basis of the famous Buchberger's algorithm, which produces the Groebner bases for the nonzero ideal II = ff 1, ff 2,, ff ss. The Buchberger's algorithm is shown in Fig. 3 [13]. Figure 3. Buchberger's Algorithm: returns a Groebner basis of II = ff 1, ff 2,, ff nn. Note, that the most efficient computer algebra systems have routines to produce Groebner bases. An example in MATHEMATICA is shown in Fig. 4. Since the Buchberger s Algorithm is based on the Multivariable Division Algorithm, which depends on the monomial term order, the computing of Groebner basis will depend on the monomial term order, as well. In Fig.4 in»in[1]:=«we want to compute the Groebner basis with respect to the lexicographic term order with xx > yy, whilst in»in[2]:=«with respect to the lexicographic term order with yy > xx. Figure 4. Computing Groebner Bases in System MATHEMATICA.

5 Universal Journal of Computational Mathematics 1(1): 5-14, Note, that Buchberger's algorithm produces a lot of»extra«basis elements than needed (i.e. it is not optimal). If we require an extra condition that no term of gg ii is divisible by any LLLL(gg jj ) and in order to ensure the uniqueness of GG (provided the monomial term order is fixed) we also require that each gg ii is monic (i.e.llll(gg ii ) = 1 for all ii = 1,2,, tt), then we get the so called reduced Groebner basis. The reduced Groebner basis always exists and is unique (see e.g. [13] for the proof). The simple algorithm which produces the reduced Groebner basis beginning with any Groebner basis GG is the following: begin with GG and make all gg ii GG monic, for any gg GG, replace gg by its remainder upon division of gg by elements if GG {gg} (in the fixed monomial term order). Of course, the routines in all computer algebra systems already return the reduced Groebner basis. In Fig. 5 and 6 Groebner basis of { xx 3 + yy, xx 2 yy yy 2 } in the lexicographic term order are computed in system MATHEMATICA and SINGULAR. We see that MATHEMATICA returns { yy 2 + yy 3, yy 2 + xxyy 2, xx 2 yy yy 2, xx 3 yy}, while SINGULAR returns GG[1] = yy3 yy2, GG[2] = xxxx2 yy2, GG[3] = xx2yy yy2, GG[4] = xx3 yy. Figure 5. Output of Groebner bases in system MATHEMATICA Figure 6. Output of Groebner bases in system SINGULAR One reason to turn to more special systems than MATHEMATICA is to compute the Groebner basis in a special monomial term order or simply to reduce a polynomial (in sense of the multivariable division algorithm) in a special monomial term order. In Fig. 7 we show the Groebner basis of { xx 3 + yy, xx 2 yy yy 2 } computed in SINGULAR with respect to the weight order with weight vector (1,3). Note, that the result GG <(1,3) = {xx 3 yy, yy 2 xx 2 yy} is not the same as the Groebner basis computed in the lexicographic monomial order GG <llllll (yy >xx) = { xx 5 + xx 6, xx 3 + yy}. Figure 7. Groebner basisgg <(1,3) of {xx 3 yy, xx 2 yy + yy 2 } computed in SINGULAR 4. Groebner Bases and Nonlinear Systems of Equations

6 10 On Groebner Bases and Their Use in Solving Some Practical Problems As mentioned before, we seek solutions (aa 1, aa 2,, aa nn ) kk of polynomial system ff 1 (xx 1, xx 2,, xx nn ) = 0, ff 2 (xx 1, xx 2,, xx nn ) = 0,, ff ss (xx 1, xx 2,, xx nn ) = 0, (4.1) where kk is the algebraic closure of kk[xx 1, xx 2,, xx nn ]. The following theorem gives a criterion on existence of solutions of (4.1). For a proof, see [1]. Let GG = {gg 1, gg 2,, gg tt } be the reduced Groebner basis of ff 1, ff 2,, ff ss. There are no solutions to the system (4.1) if and only if GG = {1}. If (4.1) has finitely many solutions, we say that ff 1, ff 2,, ff ss is zero-dimensional. Concerning Groebner basis, ff 1, ff 2,, ff ss (corresponding to (4.1)) is zero-dimensional if and only if for every ii = 1,2,, nn, there exists jj {1,2,, tt} such that LLLL(gg ii ) = xx ii αα for some αα N 0 nn. Note, that if II = ff 1, ff 2,, ff ss is not zero-dimensional, one has to compute the so called primary decomposition of II, which is much more complicated then the computations presented in the following example; see [13] for more details. We want to solve the example from [6]: ff 1 = xx 2 + yyyy + xx = 0, ff 2 = zz 2 + xxxx + zz = 0, ff 3 = yy 2 + xxxx + yy = 0. (4.2) To that end, fix the term order to be lexicographic xx > yy > zz. We find Groebner basis of ff 1, ff 2, ff 3 using system MATHEMATICA (see Fig. 8): Figure 8. Groebner basis of ff 1, ff 2, ff 3 associated to (4.1). Since the first polynomial depends only on zz, zz is either 0, 1 or 1. The system has obviously finitely many 2 solutions, since the third polynomial in GG contains only zz and yy and its leading power product is yy 2. And finally, the last polynomial contains xxandyyand zzand its leading power product is xx 2. If zz = 0 the system becomes yy + yy 2 = 0, xxxx = 0, xx + xx 2 = 0. And the (reduced) set of polynomials is already a reduced Groebner basis of the ideal it generates. The corresponding solutions are yy = 0 and xx = 0 or xx = 1 and yy = 1and xx = 0. Our solutions so far are (0,0,0), ( 1,0,0) and (0, 1,0). Similar, for zz = 1 we get yy 2 = 0, xx + yy = 0, xxxx = 0, xx + xx 2 yy = 0. The corresponding reduced Groebner basis is {yy, xx}, which yields xx = yy = 0. So another solution is (0,0, 1). Similar we get for zz = 1 the corresponding reduced Groebner 2 basis: {2yy + 1,2xx + 1}, which yields the final solution ( 1, 1, 1 ) Groebner Bases and Integer Linear Programming Let aa ii,jj Z, bb ii Z and cc jj R with ii = 1,2,, nn and jj = 1,2,, mm. We seek a solution xx = (xx 1, xx 2,, xx nn ) of the system aa 11 xx 1 + aa 12 xx aa 1nn xx nn = bb 1 (5.1) aa mm1 xx 1 + aa mm2 xx aa mmmm xx nn = bb mm, which minimizes the cost function cc(xx 1, xx 2,, xx mm ) = nn jj =1 cc jj xx jj. We call (5.1) an integer (linear) program (IP) and write it in a matrix form: minimize cc xx subject to AAxx = bb, where AA Z mm nn and bb = (bb 1,, bb mm ) Z mm. We will consider just the main mathematical idea which makes use of Groebner bases when solving IP (5.1). We can associate to (5.1) new variables XX kk ; kk = 1,2,, mm to represent the kk th equation in (5.1) as: XX kk aa kk1 xx 1 +aa kk2 xx 2 + +aa kkkk xx nn = XX kk bb kk. Of course, we can then write the whole system as aa XX 11 xx 1 +aa 12 xx 2 + +aa 1nn xx nn aa 1 XX mm 1 xx 1 +aa mm 2 xx 2 + +aa mmmm xx nn mm bb = XX 1 bb 1 XX mm mm, which is equivalent to XX 1 aa 11 XX mm aa mm 1 xx1 XX 1 aa 1nn XX mm aa mmmm xxnn = XX bb. Next, to each column of (5.1) or equivalently to each term in the brackets ( ) in the above equation we associate a new aa variable YY kk = XX 1kk aa 1 XX mmmm mm ; for each kk = 1,2,, nn. The first step in solving our problem is to figure out whether a solution exists at all. The theory of Groebner bases helps to characterize the existence and optimality of IP (5.1). The main idea is connected with the following ring homomorphism Φ: kk[yy 1,, YY nn ] kk[xx 1,, XX mm ], defined by: Φ(YY kk ) = XX 1 aa 1kk XX mm aa mmmm, (5.2)

7 Universal Journal of Computational Mathematics 1(1): 5-14, yielding Φ(YY 1 xx 1 YY nn xx nn ) = XX bb. This implies (see [9] for details) the following: there exist a solution to IP (5.1) (i.e. a vector xx = xx such that AAxx = bb ) if and only if XX bb is in the image of Φ ; yielding PP such that PP = YY xx for some xx N 0 nn. Next, the basic idea of Conti & Traverso's algorithm[4] is presented. But first we have to consider how to transform (5.1) which can contain some negative integres; recall that aa ii,jj Z and bb ii Z. This can be generally transformed to an IP with strictly nonnegative (integer) coefficients aa ii,jj, bb ii by adding an extra indeterminate WWdefined by XX 1 XX 2 XX mm WW = 1, (5.3) which transforms XX 1 aa 1jj XX ii aa iiii XX mm aa mmmm XX 1 aa 1jj +aa iiii XX ii 0 XX mm aa mmmm +aa iiii WW aa iiii =: XX AAAA WW jj. If there are some negative entries in bb, we transform XX bb to XX bb WW bb in a similar way. The optimal solution of IP (5.1) with some negative integers is therefore obtained in the following way: Define WW by (5.3), if there are some negative entries in AA, bb Define an ideal II = YY 1 XX AA 1,, YY nn XX AA nn on the polynomial ring kk[xx 1,, XX mm, YY 1,, YY nn ], if there are no negative entries in AA, bb Define an ideal II = YY 1 XX AA1 WW 1,, YY nn XX AAAA WW nn, XX 1 XX2 XXmm WW 1 on the polynomial ring kk[xx 1,, XX mm, WW, YY 1,, YY nn ], if there are some negative entries in AA, bb Let GG be the reduced Groebner basis of IIwith respect to a monomial order < cc, where cc is defined by the cost to function cc xx Dividing XX bb WW bb (i.e. the generalization of XX bb ) by GGalways yields a remainder RR kk[yy 1,, YY nn ], which ensures the optimality of the solution due to its minimality (ensured by the multivariable division algorithm); thus the solution xx = (ββ 1,, ββ nn ) to IP (5.1) is obtained by reducing XX bb WW bb ββ by GG which yields a remainder RR = YY 1 ββ 1 YY nn nn and thereby the solution xx = (ββ 1,, ββ nn ). Next, we consider the example from [9]. Following (5.1), we have to minimize the cost function cc xx = 1000xx 1 + xx 2 + xx xx 4 subject to 3xx 1 2xx 2 + xx 3 xx 4 = 1 4xx 1 + xx 2 xx 3 = 5. (5.4) The solution to the above example obtained with system SINGULAR is shown in Fig.9. The weighted term order is used with CC = ( , , ,1000,1,1,100) to ensure that XX1 > XX2 > WW > YY1 > YY2 > YY3 > YY4 and to ensure the weight order (1000,1,1,100), corresponding to cc = (1000,1,1,100). Note, that for example the monomials XX bb WW bb and XX AA2 WW 2 are: XX bb WW bb = XX 1 1 XX 2 5 = XX 1 1 XX 2 1 XX 2 1 XX 2 5 = WW 1 XX 2 6, XX AA2 WW 2 = XX 1 2 XX 2 1 = XX 1 2 XX 2 2 XX 2 2 XX 2 1 = WW 2 XX 2 3. The optimal solution xx = (1,3,2,0) is obtained from the result of the multivariable division: WW 1 XX 2 6 GG YY 1 1 YY 3 2 YY3 2 0 YY4. Figure 9. Computing the optimal solution of IP (5.4) in system SINGULAR.

8 12 On Groebner Bases and Their Use in Solving Some Practical Problems 6. Groebner Bases and Computing the Chromatic Number One of the most important and applied things in graph theory is the chromatic number of a graph. It is defined as the smallest number of colours needed to colour the vertices of graph GG = (VV, EE) so that no two adjacent vertices kk, ss VV share the same colour. For many (families) of graphs the chromatic numbers are known (i.e. defined in terms of the number of its vertices and/or edges). Probably the most simple examples are cycle graphs CC nn : a cycle graph CC 2kk has chromatic number 2, whilst CC 2kk+1 has chromatic number 3, which is usually denoted by XX(CC 2kk+1 ) = 3, XX(CC 2kk ) = 2. There are many well-known conjectures and open problems concerning the chromatic number of (undirected) graphs (e.g. Hadwiger conjecture, Albertson conjecture, Erdös Faber Lovász conjecture). The subject inspired many researchers (e.g. [2,11,15]). The idea of finding a nn colloring and consequently the chromatic number of a given graph using Groebner bases is to associate a variable xx[kk] to each vertex kk of the graph and to reduce the problem to the solution of a system of polynomial equations. The nn th roots of unity is used as nn colours. Since variables xx[kk] represent the vertices, the condition that a vertex kk should have a colour is then associated to roots of applying the following algorithm: Input: Graph GG = (VV, EE) (i.e. vertices xx[1],, xx[nn] and the adjacency matrix AA(GG)) Output: chromatic number XX(GG) Procedure: nn 1 WHILE GGGGGG = {1} NN DO II = kk=1 {xx[kk] nn 1} For all adjacent vertices kk and ss compute polynomial FFFF[kk, ss] defined by (6.2) and add it to the ideal II: II: = II FFFF[kk, ss] Compute Groebner basis GGGGGG of II IF GGGGGG = {1} THEN nn nn + 1 Find a solution (i.e. colouring) of GGGGGG = {0,,0}; XX(GG) = nn As an example, the computation of XX(G) (where GG = KK 5 - a complete graph on 5 vertices; see Fig. 10) using the basic system MATHEMATICA is presented in Figs. 11 and 12. In Fig. 11 we see that GGGG3 = {1}. Similarly we obtain that GGGG4 = {1}. Note that the command Do[Print[Fn[kk, ss]], {kk, NN}, {ss, kk 1}] is very useful since it gives a list of all possible edges. xx[kk] nn 1 = 0 (6.1) and the polynomial (xx[kk] nn xx[ss] nn )/(xx[kk] xx[ss] ) is associated to the condition that the vertices kk and ss (corresponding to xx[kk] and xx[ss] ) must have different colours (see also [1]). Thus, if vertices kk and ss are adjacent and the graph has to be coloured by nn colours, the polynomial FFFF[kk, ss] = xx[kk] nn 1 + xx[kk] nn 2 xx[ss] xx[kk] 1 xx[ss] nn 2 + xx[ss] nn 1 (6.2) must vanish. Thus finding a chromatic number of a given graph GG = (VV, EE) with VV = NN is then obtained by Figure 10. Graph GG = KK 5. Figure 11. Computing GGGG3 = {1} for GG = KK 5 and nn = 3.

9 Universal Journal of Computational Mathematics 1(1): 5-14, Figure 12. Computing GGGG5 for GG = KK 5 ; nn = 5 and solving system»gi5=0«. In Fig. 12 GI5 is computed and the solution to GI5 = 0 is verified. Though it is well-known that XX(KK nn ) = nn, yet the example is very practical and instructive (since KK nn contains all (simple) edges on nn points). In package»combinatorica«it is possible to compute the chromatic number of a given graph in MATHEMATICA. However, the above procedure may be useful to handle some general (families of) graphs. 7. Groebner Bases and Systems of ODE s Concerning the qualitative analysis of systems of ODE's (e.g. solving the center-focus problem, cyclicity problems, critical period perturbations, finding linearizability and isochronicity conditions for a given polynomial family), computing of Groebner basis is the first step toward the solution of the problem. Practical problems of this kind are related to questions like: are two polynomial ideals the same, what is the radical of given ideal, etc. The system xx = AAxx + XX (xx ), where AA is a matrix and XX (xx ) represents nonlinear terms, is linearizable if there is an analytic normalizing transformation xx = yy + h (yy ), where h (yy ) represents the nonlinear terms, that places xx = AAxx + XX (xx ) into the normal form yy = AAyy. By the Hilbert Basis Theorem every ideal in the polynomial ring kk[xx 1, xx 2,, xx nn ] over a field kk is finitely generated. See [5] for the proof. Moreover, every ascending chain of ideals II 1 II 2 II 3 in kk[xx 1, xx 2,, xx nn ] stabilizes, which means that there exists mm 1 such that for every jj > mm, II jj = II mm (see [13] for the proof). This is the main idea behind the qualitative investigation of dynamics in polynomial systems of ODE s. II = aa aa 01 bb 10 + bb 11 2, aa aa 2 01 bb aa 01 bb 10 bb 11 + bb 11 Among many problems we show an original result from [12]. In particular, the problem is arriving from the following 3D system uu = vv + aauu 2 + aavv 2 + cccccc + dddddd, vv = uu + bbuu 2 + bbvv 2 + eeeeee + ffffff, ww = ww + SSuu 2 + SSvv 2 + TTTTTT + UUUUUU, (7.1) where aa, bb, cc, dd, ee, ff, SS, TT, UU are real coefficients. The system (7.1) was already studied in [7,12] and [8] where planar polynomial systems of ODE's appearing on the center manifold of (7.1) were investigated. Often in order to consider the dynamics on a 2D center manifold of a 3D system like (7.1); i.e. in order to consider a system of the form uu = vv + (aa + dddd)(uu 2 + vv 2 ), vv = uu + (bb dddd)(uu 2 + vv 2 ) (7.2) one has to introduce the following complex coordinates xx = uu + iiii and yy = uu iiii. Then (7.2) after substitution aa 11 = bb 11 = dd, aa 01 = bb + iiii, bb 10 = bb iiii yields the following complex system: xx = ii(xx aa 11 xx 2 yy aa 01 xxxx) yy = ii(aa + bb 11 xxxx 2 + bb 10 xxxx), (7.3) where aa kkkk, bb kkkk C. The following result is based on computing of Groebner basis GG = {bb 2 11, aa 01 bb 10 + bb 11 } (with respect to the degree lexicographic order) of the (linearizability) ideal II, which is in this particular case(see (Romanovski et.al., 2013) for details) defined by: 2 + 2aa 11 (5aa 01 bb 10 + bb 11 ). 4

10 14 On Groebner Bases and Their Use in Solving Some Practical Problems Theorem 6.1. System (7.3) is linearizable if and only if one of the following conditions holds: (i) aa 01 bb 10 + bb 11 = bb 10 = aa 11 bb 11 = 0; (ii) aa 01 bb 10 + bb 11 = aa 01 = aa 11 bb 11 = Conclusions In general, mathematical theories are considered to be more valuable if they turn out to be useful in a broader variety of fields. In order to get an idea of the value of Groebner basis, we have listed some applications. The use of Groebner bases theory in studying systems of ODE's is very wide. See for example [7,8,12,13] and the references therein. The geometrical origin of integer (linear) programming is considered in [14]. System SINGULAR (see [10] is a free computer algebra system for polynomial computations. It can be downloaded at SINGULAR features one of the fastest implementations of Buchberger s algorithm to compute a Groebner basis. We used system SINGULAR for computing Groebner bases with respect to different monomial order. Among many other applications in science and engineering we emphasize just the use of Groebner bases in coding theory and in robotics. Acknowledgments The author acknowledges the support of this work by the Slovenian Research Agency. REFERENCES [1] W.W. Adams, P. Loustaunau. An introduction to Groebner basis, AMS, Providence, RI, [2] S. Akbari, M. Aryapoor, M. Jamaali. Chromatic number and clique number of subgraphs of regular graph of matrix algebras. Linear Algebra and its Applications (2012), no. 436, p [3] B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restlasseringes nach einem nulldimensionalen Polynomideal. PhD Thesis, Mathematical Institute, University of Innsbruck, Austria, [4] P. Conti, C. Traverso. Buchberger algorithm and integer programming. Proceedings AAECC-9 (new Orleans), Springer LNCS, (1991) 539, p [5] D. Cox, J. Little, D. O'Shea. Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geomety and Commutative Algebra. New York: Springer, [6] S.R. Czapor. Groebner basis methods for solving algebraic equations. Ph.D Thesis. University of Waterloo, Canada, [7] V.F. Edneral, A. Mahdi, V.G. Romanovski, D.S. Shafer. The center problem on a center manifold in R3, Nonlinear Anal., (2012) Vol. 75, p [8] B. Ferčec, M. Mencinger. Isochronicity of centers at a center manifold, AIP conference proceedings, Melville, N.Y.: American Institute of Physics, (2012), p [9] S. Flory, E. Michel. Integer programming with Groebner bases, Online available from: rd.michel/documents/else/discreteoptimization.pdf [10] G.M. Greuel, G. Pfister, H. Schönemann. Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, [11] E.L. Lawler. A note on the complexity of the chromatic number problem. Information Processing Lett., (1976), Vol. 5, No. 3, p [12] V.G. Romanovski, M. Mencinger, B. Ferčec. Investigation of center manifolds of 3-dimsystems using computer algebra. Program. comput. softw., (2013), Vol. 39, No. 2, p [13] V.G. Romanovski, D. Shafer. The center and cyclicity problems: A computational algebra approach. Boston: Birkhauser Verlag, [14] R.R. Thomas. A Geometric Buchberger Algorithm for Integer Programming. Mathematics of Operations Research, (1995), Vol 11, No.1, p [15] W. Wang. Total chromatic number of planar graphs with maximum degree ten. Graph Theory, (2007), Vol. 54, p