Gröbner bases talk at the ACAGM Summer School Leuven p. 1

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1 Gröbner bases talk at the ACAGM Summer School Leuven p. 1 Gröbner bases talk at the ACAGM Summer School Leuven Hans Schönemann hannes@mathematik.uni-kl.de Department of Mathematics University of Kaiserslautern

2 Gröbner bases talk at the ACAGM Summer School Leuven p. 2 Motivation solving the ideal membership problem for polynomial rings computing the Hilbert function, dimension, etc. for ideal in polynomial rings solving of polynomial equations

3 Gröbner bases talk at the ACAGM Summer School Leuven p. 3 What is a Gröbner basis? Let K be a field, R = K[x] the ring polynomials over K with indeterminants x 1,...,x n. Ideal generated by f 1,...,f k : I := f 1,...,f k = { k h i f i,h i polynomials } i=1

4 Gröbner bases talk at the ACAGM Summer School Leuven p. 3 What is a Gröbner basis? Let K be a field, R = K[x] the ring polynomials over K with indeterminants x 1,...,x n. Ideal generated by f 1,...,f k : I := f 1,...,f k = { k h i f i,h i polynomials } i=1 G = {g 1,...,g s } is a Gröbner basis of I, if f I NF(f G) = 0

5 Gröbner bases talk at the ACAGM Summer School Leuven p. 4 The Geometry-Algebra Dictionary Ideals in Polynomial Rings We always work over a field K. Consider the polynomial ring R = K[x 1,...,x n ]. If T R is any subset, all linear combinations g 1 f 1 + +g r f r, with g 1,...g r R and f r T, form an ideal T of R, called the ideal generated by T. We also say that T is a set of generators for the ideal. Hilbert s Basis Theorem Every ideal of the polynomial ring K[x 1,...,x n ] has a finite set of generators.

6 Gröbner bases talk at the ACAGM Summer School Leuven p. 5 The Geometry-Algebra Dictionary Algebraic Sets I The affine n-space over K is the set A n (K) = { (a 1,...,a n ) a 1,...,a n K }. Definition. If T R is any set of polynomials, its vanishing locus in A n (K) is the set V(T) = {p A n (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 the ideal T generated by T. So every algebraic set in A n (K) is of type V(I) for some ideal I of R. By Hilbert s basis theorem, it is the vanishing locus of a set of finitely many polynomials.

7 Gröbner bases talk at the ACAGM Summer School Leuven p. 6 The Geometry-Algebra Dictionary Algebraic Sets II The vanishing locus of a single non-constant polynomial is called a hypersurface of A n (K). According to our definitions, every algebraic set is the intersection of finitely many hypersurfaces. Example. The twisted cubic curve in A 3 (R) is obtained by intersecting the hypersurfaces V(y x 2 ) and V(xy z):

8 Gröbner bases talk at the ACAGM Summer School Leuven p. 7 The Geometry-Algebra Dictionary Algebraic Sets III Taking vanishing loci defines a map V which sends sets of polynomials to algebraic sets. We summarize the properties of V : Proposition. (i) The map V reverses inclusions: If I J are subsets of R, then V(I) V(J). (ii) Affine space and the empty set are algebraic: V(0) = A n (K). V(1) =. (iii) The union of finitely many algebraic sets is algebraic: If I 1,...,I s are ideals of R, then s k=1 V(I k ) = V( s k=1 I k ).

9 Gröbner bases talk at the ACAGM Summer School Leuven p. 8 The Geometry-Algebra Dictionary Algebraic Sets IV (iv) The intersection of any family of algebraic sets is algebraic: If {I λ } is a family of ideals of R, then V(I λ ) = V λ ( ) I λ. (v) A single point is algebraic: If a 1,...,a n K, then V(x 1 a 1,...,x n a n ) = {(a 1,...,a n )}. λ Computational Problem intersections. Give an algorithm for computing ideal

10 Gröbner bases talk at the ACAGM Summer School Leuven p. 9 Gröbner Bases The key idea behind Gröbner bases is to reduce problems concerning arbitrary ideals in polynomial rings to problems concerning monomial ideals.

11 Gröbner bases talk at the ACAGM Summer School Leuven p. 10 Monomial ordering monomial ordering (term ordering) on K[x 1,...,x n ]: a total ordering < on {x α α N n } with x α < x β implies x γ x α < x γ x β for any γ N n. wellordering: 1 is the smallest monomial. Let a 1,...,a k be the rows of A GL(n,R), then x α < x β if and only if there is an i with a j α = a j β for j < i and a i α < a i β. degree ordering: given by a matrix with coefficients of the first row either 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 x r+1,...,x n : L(g) K[x 1,...,x r ] implies g K[x 1,...,x r ]).

12 Gröbner bases talk at the ACAGM Summer School Leuven p. 11 What is a Gröbner basis? monomial ordering on {x α xα n n } is well ordering

13 Gröbner bases talk at the ACAGM Summer School Leuven p. 11 What is a Gröbner basis? monomial ordering on {x α xα n n } is well ordering x α x α n n > x β xβ n n if lexicographical ordering: α j = β j if j k 1 and α k > β k degree-lexicographical ordering: α i > β i or α i = β i and α j = β j if j k 1 and α k > β k

14 Gröbner bases talk at the ACAGM Summer School Leuven p. 11 What is a Gröbner basis? monomial ordering on {x α xα n n } is well ordering x α x α n n > x β xβ n n if lexicographical ordering: α j = β j if j k 1 and α k > β k degree-lexicographical ordering: α i > β i or α i = β i and α j = β j if j k 1 and α k > β k deg-lex: 77y 3 +5xy +y 2 +x+3y +1 lex: 5xy +x+77y 3 +y 2 +3y +1

15 Gröbner bases talk at the ACAGM Summer School Leuven p. 12 What is a Gröbner basis? NF(x 3 y +xy +z 2 {x 3 +z,z 2 z}) = xy yz +z

16 Gröbner bases talk at the ACAGM Summer School Leuven p. 12 What is a Gröbner basis? NF(x 3 y +xy +z 2 {x 3 +z,z 2 z}) = xy yz +z x 3 y +xy +z 2 y (x 3 +z) = xy yz +z 2 xy yz +z 2 (z 2 z) = xy yz +z

17 Gröbner bases talk at the ACAGM Summer School Leuven p. 12 What is a Gröbner basis? NF(x 3 y +xy +z 2 {x 3 +z,z 2 z}) = xy yz +z x 3 y +xy +z 2 y (x 3 +z) = xy yz +z 2 xy yz +z 2 (z 2 z) = xy yz +z normal form of f with respect to G = {f 1,...,f k }: NF(f G) h := f while ( monomial m,l(h) = ml(f i ) for some i) h := h c(h) c(f i ) mf i return (c(h)l(h)+nf(h c(h)l(h) G)

18 Gröbner bases talk at the ACAGM Summer School Leuven p. 13 Buchbergers Algorithm input: S={f 1,...f r } polynomials in K[x 1,...,x n ], < well-ordering L:={(f,g), f,g S} while L take (f,g) L, L:=L \{(f,g)} h:=nf(spoly(f,g) S) if h 0 L:=L {(h,f) f S} S:=S {(h)} end end return S

19 Gröbner bases talk at the ACAGM Summer School Leuven p. 14 Example for a Gröbner basis ideal Gröbner basis x 1 + x 2 + x 3 1 = f 1 x 1 + 2x 2 x = f 2 x 1 + 3x 3 4 = g 1 x 2 2x = g 2 NF(g 2 {f 1,f 2 }) = g 2 but NF(g 2 {g 1,g 2 }) = 0 Gröbner bases can be very complicated and their computation can take a lot of time.

20 Gröbner bases talk at the ACAGM Summer School Leuven p. 15 Standard monomials of Gröbner bases Finitely many solutions. Infinitely many solutions.

21 Gröbner bases talk at the ACAGM Summer School Leuven p. 16 Basic Properties of Gröbner bases ideal membership f I iff NF(f,GB(I))=0

22 Gröbner bases talk at the ACAGM Summer School Leuven p. 16 Basic Properties of Gröbner bases ideal membership f I iff NF(f,GB(I))=0 elimination < an elimination order for y 1,...,y n, R = K[x 1,...,x r,y 1,...,y n ]. Then GB(I) K[x 1,...,x r ] = GB(I K[x 1,...,x r ]).

23 Gröbner bases talk at the ACAGM Summer School Leuven p. 16 Basic Properties of Gröbner bases ideal membership f I iff NF(f,GB(I))=0 elimination < an elimination order for y 1,...,y n, R = K[x 1,...,x r,y 1,...,y n ]. Then GB(I) K[x 1,...,x r ] = GB(I K[x 1,...,x r ]). Hilbert function H(I)=H(L(I))

24 Gröbner bases talk at the ACAGM Summer School Leuven p. 17 Leading ideal and dimension leading ideal: L(I) = {L(f) f I} The leading monomials of a Gröbner basis generate the leading ideal. Many invariants of the leading ideal can be computed combinatorically. The ideal and its leading ideal have many common properties.

25 Gröbner bases talk at the ACAGM Summer School Leuven p. 17 Leading ideal and dimension leading ideal: L(I) = {L(f) f I} The leading monomials of a Gröbner basis generate the leading ideal. Many invariants of the leading ideal can be computed combinatorically. The ideal and its leading ideal have many common properties. the dimension the Hilbert function

26 Gröbner bases talk at the ACAGM Summer School Leuven p. 17 Leading ideal and dimension leading ideal: L(I) = {L(f) f I} The leading monomials of a Gröbner basis generate the leading ideal. Many invariants of the leading ideal can be computed combinatorically. The ideal and its leading ideal have many common properties. the dimension the Hilbert function {y 3 +x 2,x 2 y xy 2,x 4 +x 3 } is a Gröbner basis of I =< y 3 +x 2,y 4 +xy 2 > therefore L(I) =< y 3,x 2 y,x 4 > and dim C C[x,y]/I = 8

27 Gröbner bases talk at the ACAGM Summer School Leuven p. 18 Geometry of Elimination Definition. Let A A n (K) and B A m (K) be (nonempty) algebraic sets. A map ϕ : A B is a polynomial map, or a morphism, if its components are polynomial functions on A. That is, there exist polynomials f 1,...,f m R such that ϕ(p) = (f 1 (p),...,f m (p)) for all p A. The image of a morphism needs not be an algebraic set. Example. Let π : A 2 (R) A 1 (R), (a,b) b, be projection of the xy-plane onto the y-axis. Then π maps the hyperbola C = V(xy 1) onto the punctured line π(c) = A 1 (R)\{0} which is not an algebraic set.

28 Gröbner bases talk at the ACAGM Summer School Leuven p. 19 Elimination lexikographical ordering x α x α n n > x β xβ n n if α j = β j for j k 1 and α k > β k

29 Gröbner bases talk at the ACAGM Summer School Leuven p. 19 Elimination lexikographical ordering x α x α n n > x β xβ n n if α j = β j for j k 1 and α k > β k I K[x 1,...,x] ideal, G Gröbner basis, then G K[x k,...,x n ] is a Gröbner basis of I K[x k,...,x n ]. this means geometrically to compute the projection π : V(I) K n K n k+1.

30 Gröbner bases talk at the ACAGM Summer School Leuven p. 20 Solving triangular sets T = {h 1 (x 1 ),h 2 (x 1,x 2 ),...,h n (x 1,...,x n )} zero sets of triangular sets are easy to compute.

31 Gröbner bases talk at the ACAGM Summer School Leuven p. 20 Solving triangular sets T = {h 1 (x 1 ),h 2 (x 1,x 2 ),...,h n (x 1,...,x n )} zero sets of triangular sets are easy to compute. Given f 1...,f m and using Gröbner bases one can compute a given set of triangular sets T 1,...,T r such that the zero set V(f 1...,f m ) = V(T i ) is the union of the zero sets of the T i. If {1} is the minimal Göbner basis of f 1...,f m, the system has no zeros (even in no extension of the base field).

32 Gröbner bases talk at the ACAGM Summer School Leuven p. 21 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[3]=y2-y-z2+z J[2]=2yz2+z4-z2 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

33 Gröbner bases talk at the ACAGM Summer School Leuven p. 22 Solving solve(i,6); //-> [1]: [2]: [3]: [4]: [5]: //-> [1]: [1]: [1]: [1]: [1]: //-> //-> [2]: [2]: [2]: [2]: [2]: //-> //-> [3]: [3]: [3]: [3]: [3]: //->

34 Gröbner bases talk at the ACAGM Summer School Leuven p. 23 Intersection of ideals Let (F) = (f 1,...,f r ) R = K[x] and (G) = (g 1,...,g r ) R. Then GB((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

35 Gröbner bases talk at the ACAGM Summer School Leuven p. 24 Radical membership Definition{f 1 f 2...f k, f α 1 1 fα fα k k I} = I is called the radical of I. Let I R = K[x 1,...,x n ], I generated by F. f I iff 1 GB(F +(yf 1) R[y]

36 Gröbner bases talk at the ACAGM Summer School Leuven p. 25 Principal ideal I=(F) is principal (i.e. has a one-element ideal basis) iff GB(F) has exactly one element.

37 Gröbner bases talk at the ACAGM Summer School Leuven p. 26 Kernel of a ring homomorphism Let Φ be an affine ring homomorphism Φ : R = K[x 1,...,x m ]/I K[y 1,...,y n ]/(g 1,...,g s ) given by f i = Φ(x i ) K[y 1,...,y n ]/(g 1,...,g s ), i = 1,...,m. Then Ker(Φ) is generated by (g 1 (y),...,g s (y), (x 1 f 1 (y)),...,(x m f m (y))) K[x 1,...,x m ] in K[x 1,...,x m ]/I.

38 Gröbner bases talk at the ACAGM Summer School Leuven p. 27 References Gert-Martin Greuel, Gerhard Pfister: A Singular Introduction to Commutative Algebra Springer Verlag, with contributions by Olaf Bachmann, Christoph Lossen, and Hans Schönemann, 2002, Second edition appeared in Wolfram Decker, Christoph Lossen: Computing in Algebraic Geometry - A Quick Start using Singular Springer Verlag, 2005, ACM zca/ Reports_on_ca/02/paper_html/paper.html