Lecture 1: Gröbner Bases and Border Bases

Transcription

1 Lecture 1: Gröbner Bases and Border Bases The Schizophrenic Lecture Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau uni-passau.de Sophus Lie Center Nordfjordeid June 15,

2 Contents 2

3 Contents 1. Gröbner Bases 2-a

4 Contents 1. Gröbner Bases 2. Border Bases 2-b

5 Contents 1. Gröbner Bases 2. Border Bases 3. Properties of GB and BB 2-c

6 Contents 1. Gröbner Bases 2. Border Bases 3. Properties of GB and BB 4. Division Algorithms 2-d

7 Contents 1. Gröbner Bases 2. Border Bases 3. Properties of GB and BB 4. Division Algorithms 5. Neighbors 2-e

8 Contents 1. Gröbner Bases 2. Border Bases 3. Properties of GB and BB 4. Division Algorithms 5. Neighbors 6. The Buchberger Criterion 2-f

9 1 Gröbner Bases Before you criticize someone you should walk a mile in their shoes. 3

10 1 Gröbner Bases Before you criticize someone you should walk a mile in their shoes. In this way, when you criticize them, 3-a

11 1 Gröbner Bases Before you criticize someone you should walk a mile in their shoes. In this way, when you criticize them, you are a mile away 3-b

12 1 Gröbner Bases Before you criticize someone you should walk a mile in their shoes. In this way, when you criticize them, you are a mile away and you have their shoes. 3-c

13 1 Gröbner Bases Before you criticize someone you should walk a mile in their shoes. In this way, when you criticize them, you are a mile away and you have their shoes. K field P = K[x 1,...,x n ] polynomial ring over K 3-d

14 1 Gröbner Bases Before you criticize someone you should walk a mile in their shoes. In this way, when you criticize them, you are a mile away and you have their shoes. K field P = K[x 1,...,x n ] polynomial ring over K T n = {x α 1 1 xα n n α i 0} monoid of terms σ term ordering on T n (complete, multiplicative well-ordering) 3-e

15 Definition of Gröbner Bases (a) Every f P \ {0} has a unique representation f = c 1 t c s t s with c i K \ {0} and t i T n such that t 1 > σ > σ t s. The term LT σ (f) = t 1 is called the leading term of f and LC σ (f) = c 1 is the leading coefficient of f. 4

16 Definition of Gröbner Bases (a) Every f P \ {0} has a unique representation f = c 1 t c s t s with c i K \ {0} and t i T n such that t 1 > σ > σ t s. The term LT σ (f) = t 1 is called the leading term of f and LC σ (f) = c 1 is the leading coefficient of f. (b) For an ideal I P, we let LT σ (I) = LT σ (f) f I \ {0} and call it the leading term ideal of I. 4-a

17 Definition of Gröbner Bases (a) Every f P \ {0} has a unique representation f = c 1 t c s t s with c i K \ {0} and t i T n such that t 1 > σ > σ t s. The term LT σ (f) = t 1 is called the leading term of f and LC σ (f) = c 1 is the leading coefficient of f. (b) For an ideal I P, we let LT σ (I) = LT σ (f) f I \ {0} and call it the leading term ideal of I. (c) A set of polynomials f 1,...,f s I is called a σ-gröbner basis of I if LT σ (I) = LT σ (f 1 ),...,LT σ (f s ). 4-b

18 2 Border Bases Given the choice between two theories, 5

19 2 Border Bases Given the choice between two theories, take the one which is funnier. 5-a

20 2 Border Bases Given the choice between two theories, take the one which is funnier. I P zero-dimensional polynomial ideal (i.e. dim K (P/I) < ) 5-b

21 2 Border Bases Given the choice between two theories, take the one which is funnier. I P zero-dimensional polynomial ideal (i.e. dim K (P/I) < ) Open Problem: Give a good definition of border bases for higher-dimensional ideals and generalize all results in this and the subsequent lectures! 5-c

22 2 Border Bases Given the choice between two theories, take the one which is funnier. I P zero-dimensional polynomial ideal (i.e. dim K (P/I) < ) Open Problem: Give a good definition of border bases for higher-dimensional ideals and generalize all results in this and the subsequent lectures! Definition 2.1 (a) A (finite) set O T n is called an order ideal if every term dividing a term in O is contained in O. 5-d

23 2 Border Bases Given the choice between two theories, take the one which is funnier. I P zero-dimensional polynomial ideal (i.e. dim K (P/I) < ) Open Problem: Give a good definition of border bases for higher-dimensional ideals and generalize all results in this and the subsequent lectures! Definition 2.1 (a) A (finite) set O T n is called an order ideal if every term dividing a term in O is contained in O. (b) Let O be an order ideal. The set O = (x 1 O x n O) \ O is called the border of O. 5-e

24 Picture of an Order Ideal and its Border 6

25 Picture of an Order Ideal and its Border term in the order ideal term in the border 6-a

26 Definition 2.2 (a) Let O = {t 1,...,t µ } be an order ideal and O = {b 1,...,b ν } its border. A set of polynomials {g 1,...,g ν } I of the form g j = b j µ c ij t i with c ij K and t i O is called an O-border prebasis of I. i=1 7

27 Definition 2.2 (a) Let O = {t 1,...,t µ } be an order ideal and O = {b 1,...,b ν } its border. A set of polynomials {g 1,...,g ν } I of the form g j = b j µ c ij t i with c ij K and t i O is called an O-border prebasis of I. (b) An O-border prebasis of I is called an O-border basis of I if the residue classes of the terms in O are a K-vector space basis of P/I. i=1 7-a

28 Definition 2.2 (a) Let O = {t 1,...,t µ } be an order ideal and O = {b 1,...,b ν } its border. A set of polynomials {g 1,...,g ν } I of the form g j = b j µ c ij t i with c ij K and t i O is called an O-border prebasis of I. (b) An O-border prebasis of I is called an O-border basis of I if the residue classes of the terms in O are a K-vector space basis of P/I. Below se shall see that, given an O-border prebasis G, the set O is always a system of generators of the K-vector space P/ G. i=1 7-b

29 Example of a Border Basis Example 2.3 Consider the oder ideal O = {1, x, y, xy} in T 2. Its border is O = {x 2, x 2 y, xy 2, y 2 }. 8

30 Example of a Border Basis Example 2.3 Consider the oder ideal O = {1, x, y, xy} in T 2. Its border is O = {x 2, x 2 y, xy 2, y 2 }. y x 8-a

31 Example of a Border Basis Example 2.3 Consider the oder ideal O = {1, x, y, xy} in T 2. Its border is O = {x 2, x 2 y, xy 2, y 2 }. y The set of polynomials G = {g 1, g 2, g 3, g 4 } where g 1 = x 2 x, g 2 = x 2 y xy, g 3 = xy 2 xy, g 4 = y 2 y is an O-border basis of I. x 8-b

32 3 Properties of GB and BB Martin s Limerick: 9

33 3 Properties of GB and BB Martin s Limerick: The list of the theorems I knew made limericks end at line two. 9-a

34 3 Properties of GB and BB Martin s Limerick: The list of the theorems I knew made limericks end at line two. Proposition 3.1 (Existence and Uniqueness of GB) (a) For every term ordering σ and every ideal I P, there exists a σ-gröbner basis of I. 9-b

35 3 Properties of GB and BB Martin s Limerick: The list of the theorems I knew made limericks end at line two. Proposition 3.1 (Existence and Uniqueness of GB) (a) For every term ordering σ and every ideal I P, there exists a σ-gröbner basis of I. (b) A σ-gröbner basis of I is a system of generators of I. 9-c

36 3 Properties of GB and BB Martin s Limerick: The list of the theorems I knew made limericks end at line two. Proposition 3.1 (Existence and Uniqueness of GB) (a) For every term ordering σ and every ideal I P, there exists a σ-gröbner basis of I. (b) A σ-gröbner basis of I is a system of generators of I. (c) For every term ordering σ, an ideal I P has a unique reduced σ-gröbner basis, i.e. a GB which is minimal, monic, and completely interreduced. 9-d

37 For border bases, we shall always use the following notation. 10

38 For border bases, we shall always use the following notation. O = {t 1,...,t µ } order ideal with border O = {b 1,...,b ν } G = {g 1,...,g ν } is an O-border prebasis, where g j = b j µ i=1 c ij t i with c ij K 10-a

39 For border bases, we shall always use the following notation. O = {t 1,...,t µ } order ideal with border O = {b 1,...,b ν } G = {g 1,...,g ν } is an O-border prebasis, where g j = b j µ i=1 c ij t i with c ij K Proposition 3.2 (Existence and Uniqueness of BB) (a) Given an order ideal O, a 0-dimensional polynomial ideal I need not have an O-border basis, even if #O = dim K (P/I). 10-b

40 For border bases, we shall always use the following notation. O = {t 1,...,t µ } order ideal with border O = {b 1,...,b ν } G = {g 1,...,g ν } is an O-border prebasis, where g j = b j µ i=1 c ij t i with c ij K Proposition 3.2 (Existence and Uniqueness of BB) (a) Given an order ideal O, a 0-dimensional polynomial ideal I need not have an O-border basis, even if #O = dim K (P/I). (b) If a 0-dimensional ideal I P has an O-border basis G then G generates I. 10-c

41 For border bases, we shall always use the following notation. O = {t 1,...,t µ } order ideal with border O = {b 1,...,b ν } G = {g 1,...,g ν } is an O-border prebasis, where g j = b j µ i=1 c ij t i with c ij K Proposition 3.2 (Existence and Uniqueness of BB) (a) Given an order ideal O, a 0-dimensional polynomial ideal I need not have an O-border basis, even if #O = dim K (P/I). (b) If a 0-dimensional ideal I P has an O-border basis G then G generates I. (c) If a 0-dimensional ideal I P has an O-border basis G then G is uniquely determined. 10-d

42 The Relation Between GB and BB Let σ be a term ordering. Then O σ (I) = T n \ LT σ (I) is an order ideal of terms. By Macaulay s Basis Theorem, the residue classes of O σ (I) form a K-basis of P/I. 11

43 The Relation Between GB and BB Let σ be a term ordering. Then O σ (I) = T n \ LT σ (I) is an order ideal of terms. By Macaulay s Basis Theorem, the residue classes of O σ (I) form a K-basis of P/I. Proposition 3.3 (Border Bases Generalize Gröbner Bases) If O is of the form T n \ LT σ (I) for some term ordering σ, then I has an O-border basis. It contains the reduced σ-gröbner basis of I. The elements of the reduced σ-gb are exactly the border basis polynomials corresponding to the corners of O, i.e. to the minimal generators of the border term ideal. 11-a

44 4 Division Algorithms What is a proof? 12

45 4 Division Algorithms What is a proof? One half percent of alcohol. 12-a

46 4 Division Algorithms What is a proof? One half percent of alcohol. Theorem 4.1 (The Division Algorithm) Let σ be a term ordering, f P, and G = (g 1,...,g ν ) P ν. Consider the following instructions: D1. Let q 1 = = q ν = 0, p = 0, and v = f. D2. Find the smallest i {1,...,ν} such that LT σ (v) is a multiple of LT σ (g i ). If such an i exists, replace q i by q i + LM σ(v) LM σ (g i ) and v by v LM σ(v) LM σ (g i ) g i. 12-b

47 D3. Repeat step D2 until there is no more i {1,...,ν} such that LT σ (v) is a multiple of LT σ (g i ). Then replace p by p + LM σ (v) and v by v LM σ (v). D4. If now v 0, start again with step D2. If v = 0, return the tuple (q 1,...,q ν ) P ν and p P. 13

48 D3. Repeat step D2 until there is no more i {1,...,ν} such that LT σ (v) is a multiple of LT σ (g i ). Then replace p by p + LM σ (v) and v by v LM σ (v). D4. If now v 0, start again with step D2. If v = 0, return the tuple (q 1,...,q ν ) P ν and p P. This is an algorithm which returns a tuple (q 1,...,q ν ) P ν and p P such that f = q 1 g q ν g ν + p such that Supp(p) LT σ (g 1 ),...,LT σ (g ν ) =, and such that LT σ (q i g i ) σ LT σ (m) if q i a

49 D3. Repeat step D2 until there is no more i {1,...,ν} such that LT σ (v) is a multiple of LT σ (g i ). Then replace p by p + LM σ (v) and v by v LM σ (v). D4. If now v 0, start again with step D2. If v = 0, return the tuple (q 1,...,q ν ) P ν and p P. This is an algorithm which returns a tuple (q 1,...,q ν ) P ν and p P such that f = q 1 g q ν g ν + p such that Supp(p) LT σ (g 1 ),...,LT σ (g ν ) =, and such that LT σ (q i g i ) σ LT σ (m) if q i 0. Definition 4.2 The element NR σ,g (f) = p is called the normal remainder of f with respect to division by G. 13-b

50 Definition 4.3 Let O 0 = O and O i = O i 1 O i 1 for i 1. For every term t T n, there is then a unique number i = ind O (t) 0 such that t O i \ O i 1. It is called the O-index of t. Theorem 4.4 (The Border Division Algorithm) Given a polynomial f, consider the following steps: B1. Let f 1 = = f ν = 0, c 1 = = c µ = 0, and h = f. B2. If h = 0, then return (f 1,...,f ν, c 1,...,c µ ) and stop. B3. If ind O (h) = 0, then find c 1,...,c µ K such that h = c 1 t c µ t µ. Return (f 1,...,f ν, c 1,...,c µ ) and stop. 14

51 B4. If ind O (h) > 0, then let h = a 1 h a s h s with a 1,...,a s K \ {0} and h 1,...,h s T n such that ind O (h 1 ) = ind O (h). Determine the smallest index i {1,...,ν} such that h 1 factors as h 1 = t b i with a term t of degree ind O (h) 1. Subtract a 1 t g i from h, add a 1 t to f i, and continue with step B2. 15

52 B4. If ind O (h) > 0, then let h = a 1 h a s h s with a 1,...,a s K \ {0} and h 1,...,h s T n such that ind O (h 1 ) = ind O (h). Determine the smallest index i {1,...,ν} such that h 1 factors as h 1 = t b i with a term t of degree ind O (h) 1. Subtract a 1 t g i from h, add a 1 t to f i, and continue with step B2. This is an algorithm that returns a tuple (f 1,...,f ν, c 1,...,c µ ) P ν K µ such that f = f 1 g f ν g ν + c 1 t c µ t µ and deg(f i ) ind O (f) 1 for all i {1,...,ν} with f i g i a

53 B4. If ind O (h) > 0, then let h = a 1 h a s h s with a 1,...,a s K \ {0} and h 1,...,h s T n such that ind O (h 1 ) = ind O (h). Determine the smallest index i {1,...,ν} such that h 1 factors as h 1 = t b i with a term t of degree ind O (h) 1. Subtract a 1 t g i from h, add a 1 t to f i, and continue with step B2. This is an algorithm that returns a tuple (f 1,...,f ν, c 1,...,c µ ) P ν K µ such that f = f 1 g f ν g ν + c 1 t c µ t µ and deg(f i ) ind O (f) 1 for all i {1,...,ν} with f i g i 0. Corollary 4.5 The residue classes of the elements of O generate the K-vector space P/I. 15-b

54 Gröbner Bases and Rewriting Systems Let σ be a term ordering on T n and G = {g 1,...,g ν } (P \ {0}) ν. 16

55 Gröbner Bases and Rewriting Systems Let σ be a term ordering on T n and G = {g 1,...,g ν } (P \ {0}) ν. Definition 4.6 (a) Let f 1, f 2 P, and suppose there exist a constant c K, a term t T n, and an index i {1,...,ν} such that f 2 = f 1 c tg i and t LT σ (g i ) / Supp(f 2 ). Then we say that f 1 reduces to f 2 in one step, and we write f 1 g i f2. 16-a

56 Gröbner Bases and Rewriting Systems Let σ be a term ordering on T n and G = {g 1,...,g ν } (P \ {0}) ν. Definition 4.6 (a) Let f 1, f 2 P, and suppose there exist a constant c K, a term t T n, and an index i {1,...,ν} such that f 2 = f 1 c tg i and t LT σ (g i ) / Supp(f 2 ). Then we say that f 1 reduces to f 2 in one step, and we write f 1 g i f2. (b) The transitive closure of the relations g 1 g s,..., is called the rewrite relation defined by G and is denoted by. G 16-b

57 Gröbner Bases and Rewriting Systems Let σ be a term ordering on T n and G = {g 1,...,g ν } (P \ {0}) ν. Definition 4.6 (a) Let f 1, f 2 P, and suppose there exist a constant c K, a term t T n, and an index i {1,...,ν} such that f 2 = f 1 c tg i and t LT σ (g i ) / Supp(f 2 ). Then we say that f 1 reduces to f 2 in one step, and we write f 1 g i f2. (b) The transitive closure of the relations g 1 g s,..., is called the rewrite relation defined by G and is denoted by. G Proposition 4.7 A set of polynomials G = {g 1,...,g ν } is a σ-gröbner basis if and only if the rewrite rule G is confluent. This means that if there are reductions f 1 G f2 and f 1 G f3 then there exist a polynomial f 4 and reductions f 2 G f4 and f 3 G f4. 16-c

58 Border Bases and Rewriting Systems Proposition 4.8 Let G = {g 1,...,g ν } be an O-border prebasis. Then G is an O-border basis if and only if the rewriting system defined by the rules b j µ c ij t i is confluent. i=1 17

59 Border Bases and Rewriting Systems Proposition 4.8 Let G = {g 1,...,g ν } be an O-border prebasis. Then G is an O-border basis if and only if the rewriting system defined by the rules b j µ c ij t i is confluent. i=1 Notice that this rewriting system is in general not terminating, i.e. not Noetherian. This means that there may be an infinite sequences of reductions f 1 G f2 G f3 G 17-a

60 5 Neighbors Give a man a fish and he will eat for a day. Teach him how to fish, and he will 18

61 5 Neighbors Give a man a fish and he will eat for a day. Teach him how to fish, and he will sit in a boat and drink beer all day. 18-a

62 5 Neighbors Give a man a fish and he will eat for a day. Teach him how to fish, and he will sit in a boat and drink beer all day. Definition 5.1 Let b i, b j O be two distinct border terms. 18-b

63 5 Neighbors Give a man a fish and he will eat for a day. Teach him how to fish, and he will sit in a boat and drink beer all day. Definition 5.1 Let b i, b j O be two distinct border terms. (a) The border terms b i and b j are called next-door neighbors if b i = x k b j for some k {1,...,n}. 18-c

64 5 Neighbors Give a man a fish and he will eat for a day. Teach him how to fish, and he will sit in a boat and drink beer all day. Definition 5.1 Let b i, b j O be two distinct border terms. (a) The border terms b i and b j are called next-door neighbors if b i = x k b j for some k {1,...,n}. (b) The border terms b i and b j are called across-the-street neighbors if x k b i = x l b j for some k, l {1,...,n}. 18-d

65 5 Neighbors Give a man a fish and he will eat for a day. Teach him how to fish, and he will sit in a boat and drink beer all day. Definition 5.1 Let b i, b j O be two distinct border terms. (a) The border terms b i and b j are called next-door neighbors if b i = x k b j for some k {1,...,n}. (b) The border terms b i and b j are called across-the-street neighbors if x k b i = x l b j for some k, l {1,...,n}. (c) The border terms b i and b j are called neighbors if they are next-door neighbors or across-the-street neighbors. 18-e

66 Example 5.2 The border of O = {1, x, y, xy} is O = {x 2, x 2 y, xy 2, y 2 }. Here the neighbor relations look as follows: (x 2, x 2 y) and (y 2, xy 2 ) are next-door neighbor pairs (x 2 y, xy 2 ) is an across-the-street neighbor pair y x 19

67 Neighbor Syzygies Definition 5.3 (a) For t, t T n, we call the pair (lcm(t, t )/t, lcm(t, t )/t ) the fundamental syzygy of (t, t ). 20

68 Neighbor Syzygies Definition 5.3 (a) For t, t T n, we call the pair (lcm(t, t )/t, lcm(t, t )/t ) the fundamental syzygy of (t, t ). (b) The fundamental syzygies of neighboring border terms are also called the neighbor syzygies. 20-a

69 Neighbor Syzygies Definition 5.3 (a) For t, t T n, we call the pair (lcm(t, t )/t, lcm(t, t )/t ) the fundamental syzygy of (t, t ). (b) The fundamental syzygies of neighboring border terms are also called the neighbor syzygies. Proposition 5.4 (a) Given a tuple of terms (t 1,...,t r ), the fundamental syzygies σ ij = (lcm(t i, t j )/t i ) e i (lcm(t i, t j )/t j ) e j such that 1 i < j r generate the syzygy module Syz P (t 1,...,t r ) = {(f 1,...,f r ) P r f 1 t f r t r = 0}. 20-b

70 Neighbor Syzygies Definition 5.3 (a) For t, t T n, we call the pair (lcm(t, t )/t, lcm(t, t )/t ) the fundamental syzygy of (t, t ). (b) The fundamental syzygies of neighboring border terms are also called the neighbor syzygies. Proposition 5.4 (a) Given a tuple of terms (t 1,...,t r ), the fundamental syzygies σ ij = (lcm(t i, t j )/t i ) e i (lcm(t i, t j )/t j ) e j such that 1 i < j r generate the syzygy module Syz P (t 1,...,t r ) = {(f 1,...,f r ) P r f 1 t f r t r = 0}. (b) The neighbor syzygies generate the module of border syzygies Syz P (b 1,...,b ν ). 20-c

71 6 The Buchberger Criterion What is higher mathematics? 21

72 6 The Buchberger Criterion What is higher mathematics? If you awake in the morning with an unknown. 21-a

73 6 The Buchberger Criterion What is higher mathematics? If you awake in the morning with an unknown. Definition 6.1 Let g i, g j G be two distinct border prebasis polynomials. Then the polynomial S ij = (lcm(b i, b j )/b i ) g i (lcm(b i, b j )/b j ) g j is called the S-polynomial of g i and g j. 21-b

74 6 The Buchberger Criterion What is higher mathematics? If you awake in the morning with an unknown. Definition 6.1 Let g i, g j G be two distinct border prebasis polynomials. Then the polynomial S ij = (lcm(b i, b j )/b i ) g i (lcm(b i, b j )/b j ) g j is called the S-polynomial of g i and g j. Theorem 6.2 (Stetter) An O-border prebasis G is an O-border basis if and only if the neighbor syzygies lift, i.e. if and only if we have NR O,G (S ij ) = 0 for all (i, j) such that (b i, b j ) is a pair of neighbors. 21-c

75 Advantages of Border Bases 22

76 Advantages of Border Bases 1. Border bases are numerically stable. If one changes the coefficicients of some polynomials generating I slightly, the border basis of I changes continuously. 22-a

77 Advantages of Border Bases 1. Border bases are numerically stable. If one changes the coefficicients of some polynomials generating I slightly, the border basis of I changes continuously. 2. Border bases preserve symmetries. There are many more order ideals O for which a given ideal I has a border basis than order ideals of the form O σ (I) = T n \ LT σ (I). Frequently, there are border bases having the same symmetries as the initial generating system. 22-b

78 Advantages of Border Bases 1. Border bases are numerically stable. If one changes the coefficicients of some polynomials generating I slightly, the border basis of I changes continuously. 2. Border bases preserve symmetries. There are many more order ideals O for which a given ideal I has a border basis than order ideals of the form O σ (I) = T n \ LT σ (I). Frequently, there are border bases having the same symmetries as the initial generating system. 3. Border bases yield an explicit moduli space. The cefficients of a border prebasis are parametrized by an affine space. The border basis scheme is defined in this affine space by explicit equations. 22-c