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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

Buchbergers Algorithm
4
Gaussian Elimination

• Example
• 2x3y4z 5 3x4y5z 2
• ? x z-14 y 11-2z

In Gröbner bases notation Input F
2x3y4z-5, 3x4y5z-2 Output G x-z14,
y2z-11
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Euclidean Algorithm

• Computes the greatest common divisor of two
  polynomials in one variable.
• Example f1 x4-12x349x2-78x4,
• f2 x5-5x45x35x2-6x have

gcd(f1,f2) x2-3x2
In Gröbner bases notation Input F
x4-12x349x2-78x40, x5-5x45x35x2-6x Output
G x2-3x2
6
Integer Programming

• Minimize the linear function
• PNDQ
• Subject to P,N,D,Q gt 0 integer and
• P5N10D25Q 117
• This problem has the unique solution
• (P,N,D,Q) (2,1,1,4)

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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

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Integer Programming (contd)

• 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)
• p17n10d5 p12n11d5 . . .
  p2n13d5
• p2ndq4 . . . p2n13dq2 p2n12d3q

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Integer Programming (contd)

• 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 .

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Polynomial Ideals

• Let F be a set of polynomials in Kx1,,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
• ltFgt p1f1 prfr fi ? F, pi ? Kx
• These are all the polynomial
  linear combinations of elements in F.

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Hilbert Basis Theorem

• Theorem Every ideal in the polynomial ring
  Kx1,,xn is finitely generated.
• This means that any ideal I has the form ltFgt
  for a finite set of polynomials F.
• Note for the 1-variable ring Kx1, every ideal
  I is principal that is, I is generated by 1
  polynomial. This is the Euclidean Algorithm.

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Examples of Ideals

• For each example we have seen,
• ltFgt ltGgt
• lt2x3y4z-5, 3x4y5z-2gt ltx-z14, yz-11gt
• ltx4-12x349x2-78x40, x5-5x45x35x2-6xgt
  ltx2-3x2gt
• ltp5-n, p10-d, p25-qgt ltp5-n, n2-d,
  d2n-q, d3-nqgt
• 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.

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Ideal Equality

• How to check that two ideals ltFgt and ltGgt are
  equal?
• need to show that each element of F is in ltGgt and
  each element of G is in ltFgt
• Coin example
• d3-nq n(p25-q) - (p20p10dd2)(p10-d)
• p25(p5-n)

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Term Orders

• A term order is a total order lt on the set
  of all monomials xa x1a1x2a2
  xnan such that
• it is multiplicative xa lt xb ? xac lt xbc
• the constant monomial is smallest, i.e.
• 1 lt xa for all a in Nn\0

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Example term orders

• In one variable, there is only one term order 1
  lt x lt x2 lt x3 lt
• For n 2, we have
• degree lexicographic order
• 1 lt x1 lt x2 lt x12 lt x1x2 lt x22 lt x13 lt x12x2 lt
  
• purely lexicographic order
• 1 lt x1 lt x12 lt x13 lt lt x2 lt x1x2 lt x12x2 lt
  

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Initial Ideal

• Every polynomial f ? Kx1,,xn has an initial
  monomial, denoted by inlt(f).
• For every ideal I of Kx1,xn the initial ideal
  of I is generated by all initial monomials of
  polynomials in I
• inlt(I) lt inlt(f) f is in I gt

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Defining Gröbner Bases

• A finite subset G of an ideal I is a Gröbner
  basis (with respect to the term order lt) if
• inlt(g) g is in G
• generates inlt(I)
• Note there are many such generating sets. For
  instance, we can add any element of I to G to get
  another Gröbner basis .

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Reduced Gröbner Bases

• A reduced Gröbner basis satisfies
• (1) For each g in G, the coeff of inlt(g) is 1
• (2) The set inlt(g) g is in G minimally
  generates inlt(I) (nothing can be removed)
• (3) No trailing term of any g in G
  lies in the initial ideal inlt(I)
• Theorem Fixing an ideal I in Kx1,,xn and a
  term order lt, there is a unique reduced Gröbner
  basis for I

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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(ltFgt).
  If G is a Gröbner Basis for
  F, then V(G) V(F)

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Hilberts 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 x2xy-10, x3xy2-25, x4xy3-70.
  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).

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Standard Monomials

• I ? Qx1,,xn an ideal, lt a term order. A
  monomial xa x1a1x2a2 xnan is standard if
  it is not in the initial ideal inlt(I).
• Example If n 3 and inlt(I) ltx13,x24,x35gt,
  the number of standard monomials is 60. If
  inlt(I) ltx13,x24, x1x34gt, then the number of
  standard monomials is infinite.

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Fundamental Theorem of Algebra

• Theorem The number of standard monomials equals
  V(I), where the zeroes are counted with
  multiplicity.
• Example F x2z-y, x2xy-yz, xz2xz-x.
  Then, using purely lex order x gt y gt z, we get
  G x2-yz-y, xyy, xz2xz-x,
  yz2yz-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-2yz2yz, y2yzy-z-3/2, z2z-1
  so that V(F) 4.

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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 inlt(I).
• Theorem dim V(I) S

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The Residue Ring

• Theorem The set of standard monomials is a
  Q-basis for the residue ring Qx1,,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 Qx1,,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 mg -
  mg where m,m are monomials of lowest degree
  s.t. m?inlt(g) m?inlt(g).
• Theorem (Buchbergers 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.

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Buchbergers Algorithm

• Input Finite list F of polynomials in
  Qx1,,xn
• Output The reduced Gröbner basis G for ltFgt.
• Step 1 Apply Buchbergers 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(mg-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.

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Termination of Algorithm

• Question Why does this loop always terminate?
  
• Step 1 Step 3
• Answer Hilberts Basis Theorem implies that
  there is no infinite ascending chain of ideals.
  Let F f1,,fd. Each nonzero p
  normalf(mf-mf) gives a strict inclusion
  ltinlt(f1),,inlt(fd)gt ? ltinlt(f1),,inlt(fd),
  inlt(p)gt . Hence the loop terminates.

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Simple Example

• Example n 1, F x23x-4,x3-5x4
• form the S-poly (Step 1)
• x(x23x-4) - 1(x3-5x4) 3x2x-4
• It has nonzero normal form p -8x8.
• 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).

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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.