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?WHAT IS... a Gröbner ? Bernd Sturmfels

A Gröbner basis is a set of multivariate polynomi- An example of a term order (for n =2) is the als that has desirable algorithmic properties. Every degree set of can be transformed into a Gröb- 1 ≺ x ≺ x ≺ x2 ≺ x x ≺ x2 ≺ x3 ≺ x2x ≺ ··· . ner basis. This process generalizes three familiar 1 2 1 1 2 2 1 1 2 techniques: Gaussian elimination for solving linear If we fix a term order ≺, then every systems of , the Euclidean algorithm for f has a unique initial term in≺(f )=xa. This is the computing the greatest common divisor of two ≺-largest monomial xa which occurs with nonzero univariate polynomials, and the Simplex Algorithm in the expansion of f. We write the terms for linear programming; see [3]. For example, the of f in ≺-decreasing order, and we often underline input for Gaussian elimination is a collection of the initial term. For instance, a quadratic polyno- linear forms such as mial is written F − − 2 2 = 2x +3y +4z 5, 3x +4y +5z 2 , f =3x2 +5x1x2 +7x1 +11x1 +13x2 +17.

and the algorithm transforms F into the Gröbner Suppose now that I is an ideal in K[x1,...,xn] . basis Then its initial ideal in≺(I) is the ideal generated G = x − z +14,y+2z − 11 . by the initial terms of all the polynomials in I: in≺(I)= in≺(f ): f ∈ I. Let K be any field, such as the real numbers G K = R, the complex numbers K = C, the rational A finite subset of I is a Gröbner basis with re- spect to the term order ≺ if the initial terms of the numbers K = Q, or a finite field K = Fp . We write elements in G suffice to generate the initial ideal: K[x1,...,xn] for the ring of polynomials in n vari- ables xi with in the field K. If F is any in≺(I)= in≺(g):g ∈G. set of polynomials, then the ideal generated by F There is no minimality requirement for being a is the set F consisting of all polynomial linear Gröbner basis. If G is a Gröbner basis for I, then combinations: any finite subset of I that contains G is also a Gröb- F = p1f1 + ···+ pr fr : f1,...,fr ∈F ner basis. To remedy this nonminimality, we say that G is a reduced Gröbner basis if and p1,...,pr ∈ K[x1,...,xn] . (1) for each g ∈G, the coefficient of in≺(g) in g is 1, In our example the set F and its Gröbner basis G (2) the set {in≺(g): g ∈G}minimally generates generate the same ideal: G = F. By Hilbert’s in≺(I), and Basis Theorem, every ideal I in K[x1,...,xn] has the (3) no trailing term of any g ∈Glies in in≺(I). form I = F; i.e., it is generated by some finite set With this definition, we have the following theorem: F of polynomials. If the term order ≺ is fixed, then every ideal I in A term order on K[x ,...,x ] is a total order ≺ 1 n K[x ,...,x ] has a unique reduced Gröbner basis. on the set of all monomials xa = xa1 ···xan which 1 n 1 n The reduced Gröbner basis G can be computed has the following two properties: from any generating set of I by a method that was (1) It is multiplicative; i.e., xa ≺ xb implies introduced in Bruno Buchberger’s 1965 dissertation. xa+c ≺ xb+c for all a, b, c ∈ Nn. Buchberger named his method after his advisor, (2) The constant monomial is the smallest; i.e., Wolfgang Gröbner. With hindsight, the idea of Gröb- 1 ≺ xa for all a ∈ Nn\{0}. ner bases can be traced back to earlier sources, in- Bernd Sturmfels is a professor of and com- cluding a paper written in 1900 by the invariant the- puter science at the University of California at Berkeley. orist Paul Gordan. But Buchberger was the first to His email address is [email protected]. give an algorithm for computing Gröbner bases.

2NOTICES OF THE AMS VOLUME 52, NUMBER 10 Gröbner bases are very useful for solving sys- form is the division algorithm. In the familar tems of polynomial equations. Suppose K ⊆ C, and case of only one x, where I = f and f let F be a finite set of polynomials in K[x1,...,xn] . has degree d, the division algorithm writes any The variety of F is the set of all common complex polynomial p ∈ K[x] as a K-linear combination of zeros: 2 d−1 1,x,x ,...,x . But the division algorithm works n G V (F)= (z1,...,zn) ∈ C : f (z1,...,zn)=0 relative to any Gröbner basis in any number of variables. ∈F for all f . How can we test whether a given set of polyno- The variety does not change if we replace F by an- mials G is a Gröbner basis or not? Consider any two  other set of polynomials that generates the same polynomials g and g in G, and form their S-poly-  −   ideal in K[x1,...,xn] . In particular, the reduced nomial m g mg . Here m and m are monomials  Gröbner basis G for the ideal F specifies the of smallest possible degree such that m ·in≺(g)=    same variety: m·in≺(g ). The S-polynomial m g − mg lies in the V (F)=V (F)=V (G)=V (G). ideal G. We apply the division algorithm with re- spect to the tentative Gröbner basis G to mg − mg. The advantage of G is that it reveals geometric The resulting normal form is a K-linear combina- properties of the variety that are not visible from tion of monomials none of which is divisible by an F . The first question that one might ask about a initial monomial from G. A necessary condition V F variety ( ) is whether it is empty. Hilbert’s Null- for G to be a Gröbner basis is stellensatz implies that    normalformG(m g − mg )=0 for all g,g ∈G. the variety V (F)isempty if and only if G equals {1}. Buchberger’s Criterion states that this necessary condition is sufficient: a set G of polynomials is a How can one count the number of zeros of a Gröbner basis if and only if all its S-polynomials given system of equations? To answer this, we have normal form zero. From this criterion, one need one more definition. Given a fixed ideal I in derives Buchberger’s Algorithm [1] for computing K[x ,...,x ] and a term order ≺ , a monomial 1 n the reduced Gröbner basis G from any given input xa = xa1 ···xan is called standard if it is not in the 1 n set F. initial ideal in≺(I). The number of standard mono- In summary, Gröbner bases and the Buchberger mials is finite if and only if every variable xi appears to some power in the initial ideal. For example, if Algorithm for finding them are fundamental no- 3 4 5 tions in algebra. They furnish the engine for more in≺(I)= x1,x2,x3 , then there are sixty standard 3 4 4 advanced computations in , monomials, but if in≺(I)= x1,x2,x1x3 , then the set of standard monomials is infinite. such as elimination theory, computing cohomology, The variety V (I) is finite if and only if the set resolving singularities, etc. Given that polynomial of standard monomials is finite, and the number models are ubiquitous across the sciences and en- of standard monomials equals the cardinality of gineering, Gröbner bases have been used by re- V (I), when zeros are counted with multiplicity. searchers in optimization, coding, robotics, control For n =1this is the Fundamental Theorem of Al- theory, statistics, molecular biology, and many gebra, which states that the variety V (f ) of a uni- other fields. We invite the reader to experiment with variate polynomial f ∈ K[x] of degree d consists of one of the many implementations of Buchberger’s { } d complex numbers. Here the singleton f is a algorithm (e.g., in CoCoA, Macaulay2, Magma, Maple, Gröbner basis, and the standard monomials are Mathematica, or Singular). 1,x,x2,...,xd−1. Our criterion for deciding whether a variety is References finite generalizes to the following formula for the [1] DAVID COX, JOHN LITTLE, and DONAL O’ SHEA, Ideals, Va- dimension of a variety. Consider a subset S of the rieties and Algorithms. An Introduction to Computa- { } variables x1,...,xn such that no monomial in the tional Algebraic Geometry and Commutative Algebra, variables in S appears in in≺(I), and suppose that second ed., Undergraduate Texts in Mathematics, S has maximal cardinality among all subsets with Springer-Verlag, New York, 1997. | | this property. That maximal cardinality S equals [2] NIELS LAURITZEN, Concrete Abstract Algebra: From Num- the dimension of V (I). bers to Gröbner Bases, Cambridge University Press, The set of standard monomials is a K-vector- 2003. space basis for the residue ring K[x1,...,xn]/I. [3] BERND STURMFELS, Two Lectures on Gröbner Bases, New The image of a polynomial p modulo I can be Horizons in Undergraduate Mathematics, VMath Lec- expressed uniquely as a K-linear combination of ture Series, Mathematical Sciences Research Institute, standard monomials. This expression is the normal Berkeley, California, 2005, http://www.msri.org/ form of p. The process of computing the normal communications/vmath/special_productions/.

NOVEMBER 2005 NOTICES OF THE AMS 3