By the famous “Four Colour Theorem”, only four colours we need to colour a map so that no bordering regions have the same colour. Using polynomial equations and Gröbner bases we can determine if three colours are sufficient for a particular map.
1. Map Colouring
The celebrated Four Colours Theorem (1) states that all maps, whether in a plane or sphere, can be coloured with four colours without neighbouring regions being the same colour. It is very easy to produce examples of maps that cannot be coloured with three colours and others that can. A way to decide if three colours are sufficient to colour one particular map, is to analyse a polynomial system associated to the map. Each colour is represented by a complex cubic root of the unit and each region by a variable such that it can assume only one of three values, that is, one of the three colours. Thus, we have for each region.
For regions and , we have . If and are neighbouring regions, and as neighbouring regions cannot have the same colour, then , that is, . In this way, a map with regions can be colored with three colours if, and only if, the system of polynomial equations
where and and represent adjacent regions, has at least one solution.
We consider the map:
Can it be colored with three colours?
To answer this question we must verify that the polynomial system 1 with and
can be solved.
2. Systems of polynomial equations
The answer of many problems in mathematics is due to the solution of systems of polynomial equations, which is not always easy. If the system is given by linear equations, we can, through Gaussian elimination, substitute the system for another equivalent whose solution is simpler. In the case of polynomial systems, it has an analogous method that we describe below.
We will denote by the set of polynomials with complex coefficients with variables . Given a set of polynomials in , we determine the ideal generated for them as being
A theorem due to the mathematician David Hilbert
It is very simple to verify that two systems of equations and , where , satisfying the condition
have the same solution.
Therefore, the strategy to solve a system consists to find polynomials which satisfy the condition 2 and with which it is possible to test if a given belongs or not to the ideal and whose common solution is easier to compute than the original system.
The existence of such was discovered theoretically during the first half of the last century by the mathematician Wolfgang Gröbner. Some years later, his student Bruno Buchberger created an algorithm to compute it. These sets are called Gröbner bases and Buchberger’s algorithm to compute them is one of the most important tools of algebraic computation.
If the ideal is of the form , we have if and only if , that is, is divisible by . Analogously, if and only if , which suggests something similar for the division of by . In fact, we can divide a polynomial in several variables by a finite set of polynomials, provided that the monomials are ordered in some way.
3. Gröbner Bases
A monomial of is an element of the form , where non-negative integers. The monomial will be denoted by . Among the orders on the monomials of , the ones that satisfy and whenever are the ones which make it possible to put into practice the algorithm of the division.
An example is the lexicographical order , where , if there exists such that and for all .
Fixed an order on the monomials, the greatest monomial of a polynomial it is called the leading term of and denoted by .
A Gröbner basis of an ideal with respect to a given monomial order, is, by definition, any set of elements of such that the leading term of any element of is divisible by the leading term of some element of . We can show that any Gröbner basis of is a set that satisfies the condition 2.
Therefore, if and only if there exists a Gröbner basis of containing a non-zero constant, since , for all monomial of .
For example, is a Gröbner basis of , with respect to any monomial order, whereas is not a Gröbner basis of with respect to the lexicographical order, because is not divisible by .
To get a Gröbner basis of an ideal, with respect to a chosen monomial order, you can apply Buchberger’s algorithm (3). We find algorithms to calculate Gröbner bases in a majority of computer algebra systems.
For instance, a Gröbner basis for the ideal , with respect to the lexicographical order is .
4. Solution of the problem and other applications
The system given in Example 1, has at most a finite number of solutions (nine variables, each of them being able to take one of three values). The problem is to determine if the system admits any solution, and in that case, to determine it.
Applying Buchberger’s algorithm to the system of Example 1, we get the following Gröbner basis , with respect to the lexicographical order:
Therefore, and,consequently, the system admits solutions. The equations
translate the following fact: we can use any colour for , , and .
As the colours are represented by the complex cubical roots of the unit, the only solution for , and is to attribute values (colours) distinct for , and , and thus, . Finally, gives us two
possibilities: or , that is, , and take distinct colours or . And thus, we find, up to permutations of the colours, all the possible solutions:
Gröbner bases also can be used to operate with ideals, converting parametric equations to their implicit equations (some surfaces and curves for examples) into nonparametric equations, to compute the minimum polynomial of algebraic numbers, to verify theorems of Euclidean Geometry, to validate constructions with Origami and to solve Sudoku puzzles (See (4), (5) and (6)).
(4) Adams, W. and Loustaunau, P., An Introduction to Gröbner Basis, AMS, Providence RI (1994).
(5) Cox, D; Little, J. and O’Shea, D., Ideals, Varieties and Algorithms, 2nd edition, Springer-Verlag, New York, (1996).
(6) Hernandes, M. E., Um Primeiro Contato com Bases de Gröbner, 28°. Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, (2011).