1. Gröbner Bases Bernd Sturmfels Mathematics and Computer Science
University of California at Berkeley

2. Gröbner Bases Method for computing with multivariate polynomials
Generalizes well-known algorithms:
Gaussian Elimination
Euclidean Algorithm (for computing gcd)
Simplex Algorithm (linear programming)

3. General Setup Set of input polynomials F = {f1,…,fn}
Set of output polynomials G = {g1,…,gm}
Information about F easier to understand through inspection of G
Buchberger’s Algorithm

4. Gaussian Elimination Example: 2x+3y+4z = 5 & 3x+4y+5z = 2
 x = z & y = 11-2z
In Gröbner bases notation
Input: F = {2x+3y+4z-5, 3x+4y+5z-2}
Output: G = {x-z+14, y+2z-11}

5. Euclidean Algorithm
Computes the greatest common divisor of two polynomials in one variable.
Example: f1 = x4-12x3+49x2-78x+4,
f2 = x5-5x4+5x3+5x2-6x have
gcd(f1,f2) = x2-3x+2
In Gröbner bases notation
Input: F = {x4-12x3+49x2-78x+40, x5-5x4+5x3+5x2-6x}
Output: G = {x2-3x+2}

6. Integer Programming: Minimize the linear function P+N+D+Q
Subject to P,N,D,Q > 0 integer and
P+5N+10D+25Q = 117
This problem has the unique solution
(P,N,D,Q) = (2,1,1,4)

7. Integer Programming and Gröbner Bases
Represent a collection C of coins by a monomial panbdcqd in the variables p,n,d,q.
E.g., 2 pennies and 4 dimes is p2d4
Input set F = {p5-n, p10-d, p25-q}
Represents the basic relationships among coins
Output set G = {p5-n, n2-d, d2n-q, d3-nq}
Expresses a more useful set of replacement rules. E.g., the expression d3-nq translates to: replace 3 dimes with a nickel and a quarter

8. Integer Programming (cont’d)
Given a collection C of coins, we use rules encoded by G to transform (in any order) C into a set of coins C’ with equal monetary value but smaller number of elements
Example (solving previous integer program):
p17n10d p12n11d p2n13d5
p2ndq p2n13dq p2n12d3q

9. Integer Programming (cont’d)
Gröbner Bases give a method of transforming a feasible solution using local moves into a global optimum.
This transformation is analogous to running the Simplex Algorithm
Now, the general theory ….

10. Polynomial Ideals Let F be a set of polynomials in K[x1,…,xn].
Here K is a field, e.g. the rationals Q, the real numbers R, or the complex numbers C.
The ideal generated by F is
<F> = { p1f1+ ··· + prfr | fi  F, pi  K[x] }
These are all the polynomial linear combinations of elements in F.

11. Hilbert Basis Theorem
Theorem: Every ideal in the polynomial ring K[x1,…,xn] is finitely generated.
This means that any ideal I has the form <F> for a finite set of polynomials F.
Note: for the 1-variable ring K[x1], every ideal I is principal; that is, I is generated by 1 polynomial. This is the Euclidean Algorithm.

12. Examples of Ideals For each example we have seen,
<F> = <G>
<{2x+3y+4z-5, 3x+4y+5z-2}> = <{x-z+14, y+z-11}>
<{x4-12x3+49x2-78x+40, x5-5x4+5x3+5x2-6x}> = <{x2-3x+2}>
<{p5-n, p10-d, p25-q}> = <{p5-n, n2-d, d2n-q, d3-nq}>
In each example, the polynomial consequences for each set (i.e. the ideal generated by them) are the same, but the elements of G reveal more structure than those of F.

13. Ideal Equality
How to check that two ideals <F> and <G> are equal?
need to show that each element of F is in <G> and each element of G is in <F>
Coin example:
d3-nq = n(p25-q) - (p20+p10d+d2)(p10-d)
+ p25(p5-n)

14. Term Orders
A term order is a total order < on the set of all monomials xa = x1a1x2a2 ··· xnan such that:
it is multiplicative: xa < xb  xa+c < xb+c
the constant monomial is smallest, i.e.
1 < xa for all a in Nn\{0}

15. Example term orders
In one variable, there is only one term order: 1 < x < x2 < x3 < ···
For n = 2, we have
degree lexicographic order
1 < x1 < x2 < x12 < x1x2 < x22 < x13 < x12x2 < ···
purely lexicographic order
1 < x1 < x12 < x13 < ··· < x2 < x1x2 < x12x2 < ···

16. Initial Ideal
Every polynomial f  K[x1,…,xn] has an initial monomial, denoted by in<(f).
For every ideal I of K[x1,…xn] the initial ideal of I is generated by all initial monomials of polynomials in I:
in<(I) = < in<(f) | f is in I >

17. Defining Gröbner Bases
A finite subset G of an ideal I is a Gröbner basis (with respect to the term order <) if
{ in<(g) | g is in G }
generates in<(I)
Note: there are many such generating sets. For instance, we can add any element of I to G to get another Gröbner basis .

18. Reduced Gröbner Bases A reduced Gröbner basis satisfies:
(1) For each g in G, the coeff of in<(g) is 1
(2) The set { in<(g) | g is in G } minimally generates in<(I) (nothing can be removed)
(3) No trailing term of any g in G lies in the initial ideal in<(I)
Theorem: Fixing an ideal I in K[x1,…,xn] and a term order <, there is a unique reduced Gröbner basis for I

19. V(F) = {(z1,…,zn)  Cn | f(z1,…,zn) = 0, f  F}
Algebraic Geometry
If F is a set of polynomials, the variety of F over the complex numbers C equals
V(F) = {(z1,…,zn)  Cn | f(z1,…,zn) = 0, f  F}
Note: The variety depends only on the ideal of F. I.e. V(F) = V(<F>) If G is a Gröbner Basis for F, then V(G) = V(F)

20. Hilbert’s Nullstellensatz
Theorem (David Hilbert, 1890): V(F) is empty if and only if G = {1}.
Easy direction: if G = {1}, then V(F) = V(G) = { }
Ex: F = {x2+xy-10, x3+xy2-25, x4+xy3-70} Here, G = {1}, so there are no common solutions. Replacing 25 above by 26, we have G = {x-2,y-3} and V(F) = V(G) = {(2,3)}.

21. Standard Monomials
I  Q[x1,…,xn] an ideal, < a term order. A monomial xa = x1a1x2a2 ··· xnan is standard if it is not in the initial ideal in<(I).
Example: If n = 3 and in<(I) = <x13,x24,x35>, the number of standard monomials is 60. If in<(I) = <x13,x24, x1x34>, then the number of standard monomials is infinite.

22. Fundamental Theorem of Algebra
Theorem: The number of standard monomials equals #V(I), where the zeroes are counted with multiplicity.
Example: F = {x2z-y, x2+xy-yz, xz2+xz-x} Then, using purely lex order x > y > z, we get G = { x2-yz-y, xy+y, xz2+xz-x, yz2+yz-y, y2-yz } Every power of z is standard, so #V(F) is infinite.
Replacing x2z-y with x2z-1 in F, we get G = { x-2yz+2y+z, y2+yz+y-z-3/2, z2+z-1 } so that #V(F) = 4.

23. Dimension of a Variety Calculating the dimension of a variety
Think of dimension intuitively: points have dimension 0, curves have dimension 1, ….
Let S  {x1,…,xn} have maximal cardinality with the property that no monomial in the variables in S appears in in<(I).
Theorem: dim V(I) = #S

24. The Residue Ring
Theorem: The set of standard monomials is a Q-basis for the residue ring Q[x1,…,xn]/I. I.e., modulo the ideal I, every polynomial f can be written uniquely as a Q-linear combination of standard monomials.
Given f, there is an algorithm (the division algorithm) that produces this representation (called the normal form of f) in Q[x1,…,xn].

25. Testing for Gröbner Bases
Question: How to test whether a set G of polynomials is a Gröbner basis?
Take g,g’ in G and form the S-polynomial m’g - mg’ where m,m’ are monomials of lowest degree s.t. m’in<(g) = min<(g’).
Theorem (Buchberger’s Criterion): G is a Gröbner basis if and only if every S-polynomial formed by pairs g,g’ from G has normal form zero w.r.t. G.

26. Buchberger’s Algorithm
Input: Finite list F of polynomials in Q[x1,…,xn]
Output: The reduced Gröbner basis G for <F>.
Step 1: Apply Buchberger’s Criterion to check whether F is a Gröbner basis.
Step 2: If “yes,” then F is a GB. Go to Step 4.
Step 3: If “no,” we found p = normalf(m’g-mg’) to be nonzero. Set F = F  {p} and go to Step 1.
Step 4: Replace F by the reduced Gröbner basis G (apply “autoreduction’’) and output G.

27. Termination of Algorithm
Question: Why does this loop always terminate?
Step Step 3
Answer: Hilbert’s Basis Theorem implies that there is no infinite ascending chain of ideals. Let F = {f1,…,fd}. Each nonzero p = normalf(m’f-mf’) gives a strict inclusion: <in<(f1),…,in<(fd)>  <in<(f1),…,in<(fd), in<(p)> . Hence the loop terminates.

28. Simple Example Example: n = 1, F = {x2+3x-4,x3-5x+4}
form the S-poly (Step 1):
x(x2+3x-4) - 1(x3-5x+4) = 3x2+x-4
It has nonzero normal form p = -8x+8.
Therefore, F is not a Gröbner basis. We enlarge F by adding p (Step 3).
The new set F  {p} is a Gröbner basis.
The reduced GB is G = {x-1} (Step 4).

29. Summary
Gröbner bases and the Buchberger algorithm are fundamental in algebra
Applications include optimization, coding, robotics, statistics, bioinformatics etc…
Advanced algebraic geometry algorithms: elimination theory, computing cohomology, resolution of singularities etc…
Try it today, using Maple, Mathematica, Macaulay2, Magma, CoCoA, or SINGULAR.