Gröbner basestalk at the
ACAGM Summer School Leuven 2011-
Hans Schonemann
Department of Mathematics
University of Kaiserslautern
Grobner bases talk at the ACAGM Summer School Leuven 2011- – p. 1
Motivation
solving the ideal membership problem for polynomial rings
computing the Hilbert function, dimension, etc. for ideal inpolynomial rings
”solving” of polynomial equations
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What is a Gröbner basis?
Let K be a field, R = K[x] the ring polynomials over K withindeterminants x1, ..., xn.
Ideal generated by f1, . . . , fk :
I := 〈f1, . . . , fk〉 = {k∑
i=1
hifi, hi polynomials }
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What is a Gröbner basis?
Let K be a field, R = K[x] the ring polynomials over K withindeterminants x1, ..., xn.
Ideal generated by f1, . . . , fk :
I := 〈f1, . . . , fk〉 = {k∑
i=1
hifi, hi polynomials }
G = {g1, . . . , gs} is a Gröbner basis of I, if
f ∈ I ⇔ NF (f |G) = 0
Grobner bases talk at the ACAGM Summer School Leuven 2011- – p. 3
The Geometry-Algebra Dictionary
Ideals in Polynomial RingsWe always work over a field K.Consider the polynomial ring R = K[x1, . . . , xn].If T ⊂ R is any subset, all linear combinations g1f1 + · · ·+ grfr, withg1, . . . .gr ∈ R and fr ∈ T , form an ideal 〈T 〉 of R, called the idealgenerated by T . We also say that T is a set of generators for the ideal.Hilbert’s Basis Theorem Every ideal of the polynomial ringK[x1, . . . , xn] has a finite set of generators.
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The Geometry-Algebra Dictionary
Algebraic Sets IThe affine n-space over K is the set
An(K) =
{
(a1, . . . , an) | a1, . . . , an ∈ K}
.
Definition. If T ⊂ R is any set of polynomials, its vanishing locus inAn(K) is the set
V (T ) = {p ∈ An(K) | f(p) = 0 for all f ∈ T}.
Every such set is called an affine algebraic set .The vanishing locus of a subset T ⊂ R coincides with that of theideal 〈T 〉 generated by T . So every algebraic set in An(K) is of typeV (I) for some ideal I of R. By Hilbert’s basis theorem, it is thevanishing locus of a set of finitely many polynomials.
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The Geometry-Algebra Dictionary
Algebraic Sets IIThe vanishing locus of a single non-constant polynomial is called ahypersurface of An(K). According to our definitions, every algebraicset is the intersection of finitely many hypersurfaces.Example. The twisted cubic curve in A3(R) is obtained byintersecting the hypersurfaces V (y − x2) and V (xy − z):
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The Geometry-Algebra Dictionary
Algebraic Sets IIITaking vanishing loci defines a map V which sends sets ofpolynomials to algebraic sets. We summarize the properties of V :Proposition.
(i) The map V reverses inclusions: If I ⊂ J are subsets of R, thenV (I) ⊃ V (J).
(ii) Affine space and the empty set are algebraic:
V (0) = An(K). V (1) = ∅.
(iii) The union of finitely many algebraic sets is algebraic: IfI1, . . . , Is are ideals of R, then
s⋃
k=1
V (Ik) = V (
s⋂
k=1
Ik).
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The Geometry-Algebra Dictionary
Algebraic Sets IV
(iv) The intersection of any family of algebraic sets is algebraic: If{Iλ} is a family of ideals of R, then
⋂
λ
V (Iλ) = V
(
∑
λ
Iλ
)
.
(v) A single point is algebraic: If a1, . . . , an ∈ K, then
V (x1 − a1, . . . , xn − an) = {(a1, . . . , an)}.
Computational Problem Give an algorithm for computing idealintersections.
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Gröbner Bases
The key idea behind Gröbner bases is to reduce problemsconcerning arbitrary ideals in polynomial rings to problemsconcerning monomial ideals.
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Monomial ordering
monomial ordering (term ordering) on K[x1, . . . , xn]: a total ordering <
on {xα|α ∈ Nn} with xα < xβ implies xγxα < xγxβ for any γ ∈ Nn.wellordering : 1 is the smallest monomial.Let a1, . . . , ak be the rows of A ∈ GL(n,R), then xα < xβ if and onlyif there is an i with ajα = ajβ for j < i and aiα < aiβ.degree ordering : given by a matrix with coefficients of the first roweither all positive or all negative.L(g) leading monomial, c(g) the coefficient of L(g) in g, g = c(g)L(g)+ smaller terms with respect to <.elimination ordering for xr+1, . . . , xn: L(g) ∈ K[x1, . . . , xr] impliesg ∈ K[x1, . . . , xr]).
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What is a Gröbner basis?
monomial ordering on {xα1
1 · . . . · xαn
n } is well–ordering
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What is a Gröbner basis?
monomial ordering on {xα1
1 · . . . · xαn
n } is well–ordering
xα1
1· . . . · x
αnn > x
β1
1· . . . · x
βnn if
lexicographical ordering: αj = βj if j ≤ k − 1 and αk > βk
degree-lexicographical ordering:∑
αi >∑
βi or∑
αi =∑
βi and αj = βj ifj ≤ k − 1 and αk > βk
Grobner bases talk at the ACAGM Summer School Leuven 2011- – p. 11
What is a Gröbner basis?
monomial ordering on {xα1
1 · . . . · xαn
n } is well–ordering
xα1
1· . . . · x
αnn > x
β1
1· . . . · x
βnn if
lexicographical ordering: αj = βj if j ≤ k − 1 and αk > βk
degree-lexicographical ordering:∑
αi >∑
βi or∑
αi =∑
βi and αj = βj ifj ≤ k − 1 and αk > βk
deg-lex: 77y3 + 5xy + y2 + x+ 3y + 1
lex: 5xy + x+ 77y3 + y2 + 3y + 1
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What is a Gröbner basis?
NF (x3y + xy + z2 | {x3 + z, z2 − z}) = xy − yz + z
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What is a Gröbner basis?
NF (x3y + xy + z2 | {x3 + z, z2 − z}) = xy − yz + z
x3y + xy + z2 − y ∗ (x3 + z) = xy − yz + z2
xy − yz + z2 − (z2 − z) = xy − yz + z
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What is a Gröbner basis?
NF (x3y + xy + z2 | {x3 + z, z2 − z}) = xy − yz + z
x3y + xy + z2 − y ∗ (x3 + z) = xy − yz + z2
xy − yz + z2 − (z2 − z) = xy − yz + z
normal form of f with respect to G = {f1, . . . , fk}:NF (f ||G)
h := fwhile (∃ monomial m,L(h) = mL(fi) for some i)
h := h− c(h)c(fi)
mfi
return (c(h)L(h) +NF (h− c(h)L(h) | G)
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Buchbergers Algorithm
input: S={f1, ....fr} polynomials in K[x1, ..., xn], < well-orderingL:={(f,g), f,g ∈ S}while L 6= ∅
take (f,g) ∈ L, L:=L \{(f, g)}h:=NF(spoly(f,g) | S)if h 6=0
L:=L∪{(h, f)∀f ∈ S}S:=S∪{(h)}
endendreturn S
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Example for a Gröbner basis
ideal
x1 + x2 + x3 − 1 = f1
x1 + 2x2 − x3 + 2 = f2
Gröbner basis
x1 + 3x3 − 4 = g1
x2 − 2x3 + 3 = g2
NF (g2|{f1, f2}) = g2 but NF (g2|{g1, g2}) = 0
Gröbner bases can be very complicated and their computationcan take a lot of time.
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Standard monomials of Gröbner bases
| • • •−−••
• • • •• • • •• • • •• • • •
Finitely many soluti-ons.
| • • •−−−−
• • • •• • • •• • • •• • • •
Infinitely many soluti-ons.
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Basic Properties of Gröbner bases
ideal membershipf ∈ I iff NF(f ,GB(I))=0
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Basic Properties of Gröbner bases
ideal membershipf ∈ I iff NF(f ,GB(I))=0
elimination< an elimination order for y1, . . . , yn,R = K[x1, . . . , xr, y1, . . . , yn]. ThenGB(I) ∩K[x1, . . . , xr] = GB(I ∩K[x1, . . . , xr]).
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Basic Properties of Gröbner bases
ideal membershipf ∈ I iff NF(f ,GB(I))=0
elimination< an elimination order for y1, . . . , yn,R = K[x1, . . . , xr, y1, . . . , yn]. ThenGB(I) ∩K[x1, . . . , xr] = GB(I ∩K[x1, . . . , xr]).
Hilbert functionH(I)=H(L(I))
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Leading ideal and dimension
leading ideal: L(I) = 〈{L(f)|f ∈ I}〉The leading monomials of a Gröbner basis generate theleading ideal.
Many invariants of the leading ideal can be computedcombinatorically.
The ideal and its leading ideal have many common properties.
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Leading ideal and dimension
leading ideal: L(I) = 〈{L(f)|f ∈ I}〉The leading monomials of a Gröbner basis generate theleading ideal.
Many invariants of the leading ideal can be computedcombinatorically.
The ideal and its leading ideal have many common properties.
the dimension
the Hilbert function
Grobner bases talk at the ACAGM Summer School Leuven 2011- – p. 17
Leading ideal and dimension
leading ideal: L(I) = 〈{L(f)|f ∈ I}〉The leading monomials of a Gröbner basis generate theleading ideal.
Many invariants of the leading ideal can be computedcombinatorically.
The ideal and its leading ideal have many common properties.
the dimension
the Hilbert function
{y3 + x2, x2y − xy2, x4 + x3} is a Gröbner basis ofI =< y3 + x2, y4 + xy2 >therefore L(I) =< y3, x2y, x4 > and dimCC[x, y]/I = 8
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Geometry of Elimination
Definition. Let A ⊂ An(K) and B ⊂ Am(K) be (nonempty)algebraic sets. A map ϕ : A → B is a polynomial map , or a morphism , ifits components are polynomial functions on A. That is, there existpolynomials f1, . . . , fm ∈ R such that ϕ(p) = (f1(p), . . . , fm(p)) for allp ∈ A.The image of a morphism needs not be an algebraic set.Example. Let π : A2(R) → A1(R), (a, b) 7→ b, be projection of thexy-plane onto the y-axis. Then π maps the hyperbola C = V (xy − 1)
onto the punctured line π(C) = A1(R) \ {0} which is not an algebraicset.
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Elimination
lexikographical ordering
xα1
1· . . . · xαn
n > xβ1
1· . . . · xβn
n if αj = βj for j ≤ k − 1 and αk > βk
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Elimination
lexikographical ordering
xα1
1· . . . · xαn
n > xβ1
1· . . . · xβn
n if αj = βj for j ≤ k − 1 and αk > βk
I ⊂ K[x1, . . . , x] ideal, G Gröbner basis, then G ∩K[xk, . . . , xn]
is a Gröbner basis of I ∩K[xk, . . . , xn].
this means geometrically to compute the projectionπ : V (I) ⊂ Kn −→ Kn−k+1.
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Solving
triangular sets T = {h1(x1), h2(x1, x2), . . . , hn(x1, . . . , xn)}zero sets of triangular sets are easy to compute.
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Solving
triangular sets T = {h1(x1), h2(x1, x2), . . . , hn(x1, . . . , xn)}zero sets of triangular sets are easy to compute.
Given f1 . . . , fm and using Gröbner bases one can compute agiven set of triangular sets T1, . . . , Tr such that the zero setV (f1 . . . , fm) = ∪V (Ti) is the union of the zero sets of the Ti.
If {1} is the minimal Göbner basis of f1 . . . , fm, the system hasno zeros (even in no extension of the base field).
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Solving
ring A=0,(x,y,z),lp;
ideal I=x2+y+z-1,
x+y2+z-1,
x+y+z2-1;
ideal J=groebner(I);
J;
J[1]=z6-4z4+4z3-z2 J[2]=2yz2+z4-z2
J[3]=y2-y-z2+z J[4]=x+y+z2-1
triangL(J);
[1]: [2]:
_[1]=z4-4z2+4z-1 _[1]=z2
_[2]=2y+z2-1 _[2]=y2-y+z
_[3]=2x+z2-1 _[3]=x+y-1
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Solving
solve(I,6);
//-> [1]: [2]: [3]: [4]: [5]:
//-> [1]: [1]: [1]: [1]: [1]:
//-> 0.414214 0 -2.414214 1 0
//-> [2]: [2]: [2]: [2]: [2]:
//-> 0.414214 0 -2.414214 0 1
//-> [3]: [3]: [3]: [3]: [3]:
//-> 0.414214 1 -2.414214 0 0
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Intersection of ideals
Let (F ) = (f1, ..., fr) ⊆ R = K[x] and (G) = (g1, ..., gr) ⊆ R. ThenGB((F ) ∩ (G)) = GB(y(F ) + (1− y)(G)) ∩R in R[y].
Example
ring R=0,(x,y,z),dp;ideal F=z;ideal G=x,y;ideal K=intersect(F,G); K;//->K[1]=yz//->K[2]=xz
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Radical membership
Definition {f1f2....fk, ∀fα1
1 fα2
2 ...fαk
k ∈ I} =√I is called the radical of I.
Let I ⊆ R = K[x1, . . . , xn], I generated by F.f ∈
√I iff 1 ∈ GB(F + (yf − 1) ⊆ R[y]
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Principal ideal
I=(F) is principal (i.e. has a one-element ideal basis) iff GB(F) hasexactly one element.
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Kernel of a ring homomorphism
Let Φ be an affine ring homomorphism
Φ : R = K[x1, . . . , xm]/I −→ K[y1, . . . , yn]/(g1, . . . , gs)
given by fi = Φ(xi) ∈ K[y1, . . . , yn]/(g1, . . . , gs) , i = 1, . . . ,m .Then Ker(Φ) is generated by
(g1(y), . . . , gs(y) , (x1 − f1(y)), . . . , (xm − fm(y))) ∩K[x1, . . . , xm]
in K[x1, . . . , xm]/I .
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References
Gert-Martin Greuel, Gerhard Pfister: A Singular Introduction toCommutative Algebra Springer Verlag, with contributions byOlaf Bachmann, Christoph Lossen, and Hans Schönemann,2002, Second edition appeared in 2007.
Wolfram Decker, Christoph Lossen: Computing in AlgebraicGeometry - A Quick Start using Singular Springer Verlag, 2005,ACM 16.
http://www.mathematik.uni-kl.de/˜zca/Reports_on_ca/02/paper_html/paper.html
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