Lecture 5: Ideals of Points

Transcription

1 Lecture 5: Ideals of Points The Vanishing Lecture Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau uni-passau.de Sophus Lie Center Nordfjordeid June 18,

2 Contents 2

3 Contents 1. Affine Point Sets 2-a

4 Contents 1. Affine Point Sets 2. Computing Vanishing Ideals 2-b

5 Contents 1. Affine Point Sets 2. Computing Vanishing Ideals 3. Projective Point Sets 2-c

6 Contents 1. Affine Point Sets 2. Computing Vanishing Ideals 3. Projective Point Sets 4. Hilbert Functions of Points 2-d

7 1 Affine Point Sets Without geometry life is pointless. 3

8 1 Affine Point Sets Without geometry life is pointless. K field P = K[x 1,...,x n ] polynomial ring over K X = {p 1,...,p s } finite set of points in K n 3-a

9 1 Affine Point Sets Without geometry life is pointless. K field P = K[x 1,...,x n ] polynomial ring over K X = {p 1,...,p s } finite set of points in K n Definition 1.1 (a) The ideal I X = {f P f(p 1 ) = = f(p s ) = 0} is called the vanishing ideal of X. 3-b

10 1 Affine Point Sets Without geometry life is pointless. K field P = K[x 1,...,x n ] polynomial ring over K X = {p 1,...,p s } finite set of points in K n Definition 1.1 (a) The ideal I X = {f P f(p 1 ) = = f(p s ) = 0} is called the vanishing ideal of X. (b) The ring A = P/I X is called the coordinate ring of X. 3-c

11 The map eval : P K s given by eval(f) = (f(p 1 ),...,f(p s )) is called the evaluation at X. Clearly, I X = ker(eval). 4

12 The map eval : P K s given by eval(f) = (f(p 1 ),...,f(p s )) is called the evaluation at X. Clearly, I X = ker(eval). Proposition 1.2 The evaluation at X induces an isomorphism of K-algebras ε : P/I X = K s. In particular, I X is a 0-dimensional ideal. 4-a

13 The map eval : P K s given by eval(f) = (f(p 1 ),...,f(p s )) is called the evaluation at X. Clearly, I X = ker(eval). Proposition 1.2 The evaluation at X induces an isomorphism of K-algebras ε : P/I X = K s. In particular, I X is a 0-dimensional ideal. Proof: The map ε is clearly injective. If p i denotes the vanishing ideal of {p i } then I X = p 1 p s. Since the ideals p i are comaximal primes, the Chinese Remainder Theorem yields P/I X = P/p1 P/p s, and therefore dim K (P/I X ) = s. 4-b

14 Definition 1.3 For i = 1,...,s, an element f P such that ε( f) = e i is called a separator of p i in X. 5

15 Definition 1.3 For i = 1,...,s, an element f P such that ε( f) = e i is called a separator of p i in X. Remark 1.4 Using a set of separators {f 1,...,f s } of X, one can solve the interpolation problem: for all a 1,...,a s K, the polynomial f = a 1 f a s f s satisfies f(p i ) = a i for i = 1,...,s. 5-a

16 Definition 1.3 For i = 1,...,s, an element f P such that ε( f) = e i is called a separator of p i in X. Remark 1.4 Using a set of separators {f 1,...,f s } of X, one can solve the interpolation problem: for all a 1,...,a s K, the polynomial f = a 1 f a s f s satisfies f(p i ) = a i for i = 1,...,s. A separator of minimal degree of p i in X is called a minimal separator of p i and its degree is called the degree of p i in X and is denoted by deg X (p i ). 5-c

17 2 Computing Vanishing Ideals There are two essential strategies for success in computing vanishing idelas: 6

18 2 Computing Vanishing Ideals There are two essential strategies for success in computing vanishing idelas: 1. Never reveal your strategy a

19 2 Computing Vanishing Ideals X = {p 1,...,p s } finite set of points in K s There are two essential strategies for success in computing vanishing idelas: 1. Never reveal your strategy. 2. Goal: Compute a system of generators (Gröbner basis, border basis) of I X! 6-b

20 2 Computing Vanishing Ideals X = {p 1,...,p s } finite set of points in K s There are two essential strategies for success in computing vanishing idelas: 1. Never reveal your strategy. 2. Goal: Compute a system of generators (Gröbner basis, border basis) of I X! Given a matrix M Mat r,s (K) in row echelon form and a row v K s, we can reduce v against M by using the pivot elements of M to eliminate the corresponding entries of v from left to right. 6-c

21 The Buchberger-Möller Algorithm Let σ be a term ordering. The following algorithm computes the reduced σ-gröbner basis of I X. 7

22 The Buchberger-Möller Algorithm Let σ be a term ordering. The following algorithm computes the reduced σ-gröbner basis of I X. M1. Let G =, O =, S =, L = {1}, and let M = (m ij ) Mat 0,s (K). 7-a

23 The Buchberger-Möller Algorithm Let σ be a term ordering. The following algorithm computes the reduced σ-gröbner basis of I X. M1. Let G =, O =, S =, L = {1}, and let M = (m ij ) Mat 0,s (K). M2. If L =, return (G, O) and stop. Otherwise, choose t = min σ (L) and delete it from L. 7-b

24 The Buchberger-Möller Algorithm Let σ be a term ordering. The following algorithm computes the reduced σ-gröbner basis of I X. M1. Let G =, O =, S =, L = {1}, and let M = (m ij ) Mat 0,s (K). M2. If L =, return (G, O) and stop. Otherwise, choose t = min σ (L) and delete it from L. M3. Compute (t(p 1 ),...,t(p s )) K s and reduce it against the rows of M to obtain (v 1,...,v s ) = (t(p 1 ),...,t(p s )) i a i (m i1,...,m is ) with a i K. 7-c

25 M4. If (v 1,...,v s ) = (0,...,0), append the polynomial t i a is i to G where s i is the i th element in S. Remove from L all multiples of t. Then continue with step M2. 8

26 M4. If (v 1,...,v s ) = (0,...,0), append the polynomial t i a is i to G where s i is the i th element in S. Remove from L all multiples of t. Then continue with step M2. M5. Otherwise append (v 1,...,v s ) as a new row to M and t i a is i as a new element to S. Add t to O, and add to L those elements of {x 1 t,...,x n t} which are neither multiples of an element of L nor of LT σ (G). Continue with step M2. 8-a

27 M4. If (v 1,...,v s ) = (0,...,0), append the polynomial t i a is i to G where s i is the i th element in S. Remove from L all multiples of t. Then continue with step M2. M5. Otherwise append (v 1,...,v s ) as a new row to M and t i a is i as a new element to S. Add t to O, and add to L those elements of {x 1 t,...,x n t} which are neither multiples of an element of L nor of LT σ (G). Continue with step M2. Corollary 2.1 If we replace step M2 in the theorem by the following instruction, the algorithm also computes minimal separators of X. 8-b

28 M4. If (v 1,...,v s ) = (0,...,0), append the polynomial t i a is i to G where s i is the i th element in S. Remove from L all multiples of t. Then continue with step M2. M5. Otherwise append (v 1,...,v s ) as a new row to M and t i a is i as a new element to S. Add t to O, and add to L those elements of {x 1 t,...,x n t} which are neither multiples of an element of L nor of LT σ (G). Continue with step M2. Corollary 2.1 If we replace step M2 in the theorem by the following instruction, the algorithm also computes minimal separators of X. M2 If L = then row reduce M to a diagonal matrix and mimic these row operations on the elements of S (considered as a column vector). Next replace S by M 1 S, return (G, O, S) and stop. If L, choose t = min σ (L) and delete it from L. 8-c

29 The BB Version of the BM Algorithm Corollary 2.2 If we replace step M2 in the theorem by the following instruction, the resulting algorithm computes the O σ (I X )-border basis of I X. 9

30 The BB Version of the BM Algorithm Corollary 2.2 If we replace step M2 in the theorem by the following instruction, the resulting algorithm computes the O σ (I X )-border basis of I X. M5 Otherwise append (v 1,...,v s ) as a new row to M and t i a is i as a new element to S. Add t to O, and add to L the elements in O which are not contained in LT σ (G). Continue with step M2. 9-a

31 Example 2.3 Let σ = DegRevLex. We want to compute the O σ (I X ) border basis of the vanishing ideal of X = {(0, 0), (1, 0), (0, 1), (2, 0)}. We follow the non-trivial steps. M2. Let t = 1. M3. (v 1,...,v 4 ) = (1,...,1). M5. M = ( ) and S = O = (1) and L = {x, y}. M2. Let t = y. M3. (v 1,...,v 4 ) = (0, 0, 1, 0). M5. M = ( ) and S = O = {1, y} and L = {x, xy, y 2 }. M2. Let t = x. M3. (v 1,...,v 4 ) = (0, 1, 0, 2). 10

32 M5. M = L = {x 2, xy, y 2 }, and S = O = {1, y, x} and M2. Let t = y 2. M3. (v 1,...,v 4 ) = (0, 0, 1, 0) (0, 0, 1, 0) = (0, 0, 0, 0) M4. Let G = (y 2 y). M2. Let t = xy. M3. (v 1,...,v 4 ) = (0, 0, 0, 0) M4. Let G = (y 2 y, xy). M2. Let t = x 2. M3. (v 1,...,v 4 ) = (0, 1, 0, 4) (0, 1, 0, 2) = (0, 0, 0, 2) 11

33 M5. M = O = {1, y, x, x 2 } and and L = {x 3, x 2 y}. M2. The next term t = x 2 y goes into G. and S = {1, y, x, x 2 x} and M2. The next term t = x 3 yields the element x 3 3x 2 + 2x of G. M2. Finally we get O = {1, y, x, x 2 } and G = (y 2 y, xy, x 2 y, x 3 3x 2 + 2x). 12

34 3 Projective Point Sets RUNTIME ERROR 6D at 417A:32CF: 13

35 3 Projective Point Sets RUNTIME ERROR 6D at 417A:32CF: Incompetent user 13-a

36 3 Projective Point Sets RUNTIME ERROR 6D at 417A:32CF: Incompetent user X = {p 1,...,p s } finite set of points in P n (K) p i = (a i0 : : a in ) with a ij K P = K[x 0,...,x n ] 13-b

37 3 Projective Point Sets RUNTIME ERROR 6D at 417A:32CF: Incompetent user X = {p 1,...,p s } finite set of points in P n (K) p i = (a i0 : : a in ) with a ij K P = K[x 0,...,x n ] Goal: Compute the homogeneous vanishing ideal I X = f P f homogeneous, f(p 1 ) = = f(p s ) = 0 of X (via a reduced Gröbner basis). 13-c

38 Remark 3.1 If none of the points of X is contained in the hyperplane Z(x 0 ), we can compute I X as follows. 14

39 Remark 3.1 If none of the points of X is contained in the hyperplane Z(x 0 ), we can compute I X as follows. (a) Normalize all p i = (a i0 :... : a in ) such that a i0 = 1. Let q i = (a i1,...,a in ) be the corresponding affine point in D + (x 0 ) = A n. 14-a

40 Remark 3.1 If none of the points of X is contained in the hyperplane Z(x 0 ), we can compute I X as follows. (a) Normalize all p i = (a i0 :... : a in ) such that a i0 = 1. Let q i = (a i1,...,a in ) be the corresponding affine point in D + (x 0 ) = A n. (b) Compute a Gröbner basis G of the vanishing ideal J X of {q 1,...,q s } in K[x 1,...,x n ] with respect to a term ordering σ. 14-b

41 Remark 3.1 If none of the points of X is contained in the hyperplane Z(x 0 ), we can compute I X as follows. (a) Normalize all p i = (a i0 :... : a in ) such that a i0 = 1. Let q i = (a i1,...,a in ) be the corresponding affine point in D + (x 0 ) = A n. (b) Compute a Gröbner basis G of the vanishing ideal J X of {q 1,...,q s } in K[x 1,...,x n ] with respect to a term ordering σ. (c) Then we have I X = JX hom and the homogenization G hom of G is a Gröbner basis of I X with resepct to the projective extension of σ. 14-c

42 Remark 3.1 If none of the points of X is contained in the hyperplane Z(x 0 ), we can compute I X as follows. (a) Normalize all p i = (a i0 :... : a in ) such that a i0 = 1. Let q i = (a i1,...,a in ) be the corresponding affine point in D + (x 0 ) = A n. (b) Compute a Gröbner basis G of the vanishing ideal J X of {q 1,...,q s } in K[x 1,...,x n ] with respect to a term ordering σ. (c) Then we have I X = JX hom and the homogenization G hom of G is a Gröbner basis of I X with resepct to the projective extension of σ. Remark 3.2 If K has enough elements, we can always use a coordinate transformation and the previous remark to compute I X. But we will only get a system of generators, not a Gröbner basis. 14-d

43 The Projective BM-Algorithm Let σ be a term ordering on P. The following algorithm computes the reduced σ-gröbner basis of the homogeneous vanishing ideal I X. 15

44 The Projective BM-Algorithm Let σ be a term ordering on P. The following algorithm computes the reduced σ-gröbner basis of the homogeneous vanishing ideal I X. P1. Let G =, S =, L = {1}, d = 0, and let M = (m ij ) be a matrix over K with s columns and initially zero rows. 15-a

45 The Projective BM-Algorithm Let σ be a term ordering on P. The following algorithm computes the reduced σ-gröbner basis of the homogeneous vanishing ideal I X. P1. Let G =, S =, L = {1}, d = 0, and let M = (m ij ) be a matrix over K with s columns and initially zero rows. P2. Compute the Hilbert series of S = P/(LT σ (g) g G) and check whether HF S (i) = s for all i d. If this is true, return G and stop. Otherwise, increase d by one, let S =, let M = (m ij ) be a matrix over K with s columns and zero rows, and let L be the set of all terms in T n+1 d which are not multiples of an element LT σ (g) with g G. 15-b

46 The Projective BM-Algorithm Let σ be a term ordering on P. The following algorithm computes the reduced σ-gröbner basis of the homogeneous vanishing ideal I X. P1. Let G =, S =, L = {1}, d = 0, and let M = (m ij ) be a matrix over K with s columns and initially zero rows. P2. Compute the Hilbert series of S = P/(LT σ (g) g G) and check whether HF S (i) = s for all i d. If this is true, return G and stop. Otherwise, increase d by one, let S =, let M = (m ij ) be a matrix over K with s columns and zero rows, and let L be the set of all terms in T n+1 d which are not multiples of an element LT σ (g) with g G. P3. If L =, continue with step 2). Otherwise, choose t = min σ (L) and remove it from L. 15-c

47 P4. For i = 1,...,s, compute t(p i ) = t(c i0,...,c in ). Reduce the vector (t(p 1 ),...,t(p s )) against the rows of M to obtain (v 1,...,v s ) = (t(p 1 ),...,t(p s )) i a i (m i1,...,m is ) with a i K. 16

48 P4. For i = 1,...,s, compute t(p i ) = t(c i0,...,c in ). Reduce the vector (t(p 1 ),...,t(p s )) against the rows of M to obtain (v 1,...,v s ) = (t(p 1 ),...,t(p s )) i a i (m i1,...,m is ) with a i K. P5. If (v 1,...,v s ) = (0,...,0) then append the polynomial t i a is i to G, where s i is the i th element of the list S. Continue with step P3. 16-a

49 P4. For i = 1,...,s, compute t(p i ) = t(c i0,...,c in ). Reduce the vector (t(p 1 ),...,t(p s )) against the rows of M to obtain (v 1,...,v s ) = (t(p 1 ),...,t(p s )) i a i (m i1,...,m is ) with a i K. P5. If (v 1,...,v s ) = (0,...,0) then append the polynomial t i a is i to G, where s i is the i th element of the list S. Continue with step P3. P6. If (v 1,...,v s ) (0,...,0) then add (v 1,...,v s ) as a new row to M and t i a is i as a new element to S. Continue with step P3. 16-b

50 P4. For i = 1,...,s, compute t(p i ) = t(c i0,...,c in ). Reduce the vector (t(p 1 ),...,t(p s )) against the rows of M to obtain (v 1,...,v s ) = (t(p 1 ),...,t(p s )) i a i (m i1,...,m is ) with a i K. P5. If (v 1,...,v s ) = (0,...,0) then append the polynomial t i a is i to G, where s i is the i th element of the list S. Continue with step P3. P6. If (v 1,...,v s ) (0,...,0) then add (v 1,...,v s ) as a new row to M and t i a is i as a new element to S. Continue with step P3. A variant of this algorithm also computes the separators of X. 16-c

51 4 Hilbert Functions of Points A sparrow in your hand... 17

52 4 Hilbert Functions of Points A sparrow in your hand is better than a dove over your head. 17-a

53 4 Hilbert Functions of Points A sparrow in your hand is better than a dove over your head. 2. will probably make a mess on your wrist. 17-b

54 4 Hilbert Functions of Points A sparrow in your hand is better than a dove over your head. 2. will probably make a mess on your wrist. P = K[x 0,...,x n ] polynomial ring over a field K X = {p 1,...,p s } P n (K) finite set of points 17-c

55 4 Hilbert Functions of Points A sparrow in your hand is better than a dove over your head. 2. will probably make a mess on your wrist. P = K[x 0,...,x n ] polynomial ring over a field K X = {p 1,...,p s } P n (K) finite set of points Definition 4.1 The function HF X : Z N given by HF X (i) = dim K (P/I X ) i is called the Hilbert function of X. 17-d

56 4 Hilbert Functions of Points A sparrow in your hand is better than a dove over your head. 2. will probably make a mess on your wrist. P = K[x 0,...,x n ] polynomial ring over a field K X = {p 1,...,p s } P n (K) finite set of points Definition 4.1 The function HF X : Z N given by HF X (i) = dim K (P/I X ) i is called the Hilbert function of X. R = P/I X homogeneous coordinate ring of X 17-e

57 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: 18

58 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: (1) Compute a Gröbner basis of the homogeneous vanishing ideal I X. 18-a

59 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: (1) Compute a Gröbner basis of the homogeneous vanishing ideal I X. (2) Compute the Hilbert function of P/ LT σ (I X ). 18-b

60 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: (1) Compute a Gröbner basis of the homogeneous vanishing ideal I X. (2) Compute the Hilbert function of P/ LT σ (I X ). Remark 4.3 The Hilbert function HF X has the following properties: 18-c

61 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: (1) Compute a Gröbner basis of the homogeneous vanishing ideal I X. (2) Compute the Hilbert function of P/ LT σ (I X ). Remark 4.3 The Hilbert function HF X has the following properties: (a) HF X (0) = 1 18-d

62 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: (1) Compute a Gröbner basis of the homogeneous vanishing ideal I X. (2) Compute the Hilbert function of P/ LT σ (I X ). Remark 4.3 The Hilbert function HF X has the following properties: (a) HF X (0) = 1 (b) There is a number a X such that HF X (i) = s for i > a X. 18-e

63 Remark 4.2 The Hilbert function HF X can be computed from the coordinates of the points as follows: (1) Compute a Gröbner basis of the homogeneous vanishing ideal I X. (2) Compute the Hilbert function of P/ LT σ (I X ). Remark 4.3 The Hilbert function HF X has the following properties: (a) HF X (0) = 1 (b) There is a number a X such that HF X (i) = s for i > a X. (c) HF X (0) < HF X (1) < < HF X (a X + 1) 18-f

64 Hilbert Functions of Subsets Remark 4.4 Suppose that Y = X \ {p i }. Then HF Y has the following properties: 19

65 Hilbert Functions of Subsets Remark 4.4 Suppose that Y = X \ {p i }. Then HF Y has the following properties: (a) There is a number d such that HF Y (i) = HF X (i) for i < d and HF Y (i) = HF X (i) 1 for i d. 19-a

66 Hilbert Functions of Subsets Remark 4.4 Suppose that Y = X \ {p i }. Then HF Y has the following properties: (a) There is a number d such that HF Y (i) = HF X (i) for i < d and HF Y (i) = HF X (i) 1 for i d. (b) The number d is precisely the degree of p i in X, i.e. the degree of a minimal separator of p i in X. 19-b

67 Hilbert Functions of Subsets Remark 4.4 Suppose that Y = X \ {p i }. Then HF Y has the following properties: (a) There is a number d such that HF Y (i) = HF X (i) for i < d and HF Y (i) = HF X (i) 1 for i d. (b) The number d is precisely the degree of p i in X, i.e. the degree of a minimal separator of p i in X. (c) The number d satisfies d a X c

68 Hilbert Functions of Subsets Remark 4.4 Suppose that Y = X \ {p i }. Then HF Y has the following properties: (a) There is a number d such that HF Y (i) = HF X (i) for i < d and HF Y (i) = HF X (i) 1 for i d. (b) The number d is precisely the degree of p i in X, i.e. the degree of a minimal separator of p i in X. (c) The number d satisfies d a X + 1. (d) There exists at least one point p i in X of degree a X + 1, i.e. such that Y = X \ {p i } satisfies HF Y (i) = HF X (i) for i a X and HF Y (i) = HF X (i) 1 for i a X d

69 Corollary 4.5 For every k 1, the truncated Hilbert function h(i) = min{hf X (i), k} is the Hilbert function of at least one subset Y of X consisting of k points. 20