From Gauss. to Gröbner Bases. John Perry. The University of Southern Mississippi. From Gauss to Gröbner Bases p.

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1 From Gauss to Gröbner Bases p. From Gauss to Gröbner Bases John Perry The University of Southern Mississippi

2 From Gauss to Gröbner Bases p. Overview Questions: Common zeroes? Tool: Gaussian elimination Gröbner bases Solutions to questions Recent advances

3 From Gauss to Gröbner Bases p. Part 1: some interesting questions Given f 1, f 2,..., f m in x1, x 2,..., x n : are there common zeroes over n? if so, dimension of solution space? (variety) if zero-dimensional, how many solutions?

4 From Gauss to Gröbner Bases p. Example 1 f 1 = x f 2 = y Common zeroes in 2? If so, dimension of solution space? If zero-dimensional, how many solutions?

5 From Gauss to Gröbner Bases p. Example 1 f 1 = x f 2 = y Common zeroes in 2? If so, dimension of solution space? If zero-dimensional, how many solutions? x 2 + 1=0 y 2 + 1=0 = x=±i = y=±i (i,±i),( i,±i)

6 From Gauss to Gröbner Bases p. Example 1 f 1 = xy 2 2x y f 2 = x 2 y x 2 + y 1 Common zeroes in 2? If so, dimension of solution space? If zero-dimensional, how many solutions?

7 From Gauss to Gröbner Bases p. Motivation Mathematical applications Algebraic geometry (Cox, Little, O Shea textbooks) Algebraic statistics (Sturmfels) Automated theorem proving (various textbooks) Commutative algebra (Kreuzer, Robbiano textbooks) Differential equations and differential algebra (Lidl, Pilz textbook) Partial differential equations (Golubitsky, Hebert, Ovchinnikov)

8 From Gauss to Gröbner Bases p. Motivation Non-mathematical applications Astronomy (singularities) Coding Theory (de Boer, Pellikaan) Cryptography (cell phones, NSA, Ackerman and Kreuzer) Mathematical biology (Stigler: reverse-engineering gene networks) Military (Arnold: automatic radar targeting) Petri nets (von zur Gathen, Gerhard: parallel processes)

9 From Gauss to Gröbner Bases p. Example 1: linear f 1 = 4w 8x 12z f 3 = 2x+ 4y f 2 = 3w+ 3y 12z f 4 = 3y+ z Common zeroes in? If so, dimension of solution set? If zero-dimensional, how many solutions? Tool: Gaussian elimination

10 From Gauss to Gröbner Bases p. Example 1: linear start with inputs: 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 3w +3y 12z = 0

11 From Gauss to Gröbner Bases p. Example 1: linear identify critical pair: 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 3w +3y 12z = 0

12 From Gauss to Gröbner Bases p. Example 1: linear compute subtraction polynomial s: 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 24x +12y 12z = 0 3 f f 4 = 24x+ 12y 12z

13 From Gauss to Gröbner Bases p. Example 1: linear reduce s modulo current F: 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 24x +12y 12z = 0 3 f f 4 = 24x+ 12y 12z

14 From Gauss to Gröbner Bases p. Example 1: linear reduce s modulo current F: 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 36y 12z = 0 3 f f 4 = 12 f 2 36y 12z

15 From Gauss to Gröbner Bases p. Example 1: linear echelon form! 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 0 = 0 3 f f 4 = 12 f 2 12 f 3

16 From Gauss to Gröbner Bases p. Example 1: linear echelon form! 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 0 = 0 3 f f 4 = 12 f 2 12 f 3 Echelon form: Infinitely many zeroes One-dimensional set of zeroes ( 13s, 2s, s, 3s), s

17 From Gauss to Gröbner Bases p. 1 Summary so far Given a linear system... Gaussian elimination nice form nice form info: zeroes What about non-linear systems???

18 From Gauss to Gröbner Bases p. 1 Aim Linear: Gaussian elimination echelon form Non-linear: Buchberger s algorithm Gröbner basis

19 From Gauss to Gröbner Bases p. 1 Example 2: nonlinear f 1 = xy 2 2x y f 2 = x 2 y x 2 + y 1 Common zeroes in 2? Yes If so, dimension? 0 If zero-dimensional, how many? 5 Nice form and its application are not obvious!

20 From Gauss to Gröbner Bases p. 1 Part 2: Gröbner bases How can we find (and use) this nice form?

21 From Gauss to Gröbner Bases p. 1 Goal Algorithm Input: Output: Generalized Gaussian elimination F= f 1, f 2,..., f m G= g 1, g 2,..., g r, a nice form of F

22 From Gauss to Gröbner Bases p. 1 Goal Algorithm Generalized Gaussian elimination Input: F= f 1, f 2,..., f m Output: G= g 1, g 2,..., g r, a nice form of F 1. G=F 2. Identify critical pairs p, q from pivots 3. Loop through critical pairs: (a) s =σ p p σ q q (b) reduce s modulo G, append reduced to G 4. Nice form? YES: Done! NO: go to 2

23 From Gauss to Gröbner Bases p. 1 Generalize to non-linear: Pivots? Elimination? Reduction? Nice form?

24 From Gauss to Gröbner Bases p. 1 Pivots? f 1 = x 2 + xy+ y 2 f 2 = x z+ z 3 For f 1 : x 2 + xy+ y 2 x 2 + xy+ y 2 x 2 + xy+ y 2 For f 2 : x z+ z 3 x z+ z 3

25 From Gauss to Gröbner Bases p. 1 Pivots? Leading terms! Notation: leading term lt ( f) Term ordering ( ) admissible iff t, u, v 1. u v or v u or u= v 2. u v implies v u or v=u 3. u v implies t u t v

26 From Gauss to Gröbner Bases p. 1 Leading terms, examples lex(x y z): f 1 = x 2 + xy+ y 2 f 2 = x z+ z 3 x 2 + xy+ y 2 x z+ z 3 tdeg(x y z): x 2 + xy+ y 2 x z+ z 3

27 From Gauss to Gröbner Bases p. 1 Term orderings: notes { } m n lex tdeg Use: multiply matrix by exponent vector to obtain weight vector, compare top to bottom

28 Term orderings: examples Compare x, y z using lex: = = So x lex y z. From Gauss to Gröbner Bases p. 2

29 Term orderings: examples Compare x, y z using tdeg: = = So x tdeg y z. From Gauss to Gröbner Bases p. 2

30 From Gauss to Gröbner Bases p. 2 Term orderings: notes Often useful to change To answer our questions, any ok To find explicit zeroes, need lex

31 From Gauss to Gröbner Bases p. 2 Elimination? S-polynomials! Multiply to lcm, subtract: Thus the S-polynomial f 1 = x f 2 = y y 2 x x 2 y y 2 x 2 S ( f 1, f 2 )= y 2 x 2

32 From Gauss to Gröbner Bases p. 2 Reduction? linear case Recall 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 24x +12y 12z = 0 4w 8x 12z = 0 2x +4y = 0 3y +z = 0 0 = 0 3 f 1 4 f 4 = 12 f f 3

33 From Gauss to Gröbner Bases p. 2 Reduction? non-linear case S ( f 1, f 2 ) = y 2 x 2 f 1 = x f 2 = y 2 + 1

34 From Gauss to Gröbner Bases p. 2 Reduction? non-linear case f 1 = x f 2 = y S ( f 1, f 2 ) = 1 f f 2

35 From Gauss to Gröbner Bases p. 2 Representation S ( f 1, f 2 ) has a representation modulo f 1,f 2,...,f m if h 1, h 2,..., h m s.t. S ( f 1, f 2 )= h 1 f 1 + h 2 f h m f m h k s obtainable by reduction: k= 1,..., m h k 0 implies lt hk lt fk lcm lt ( f 1 ),lt ( f 2 )

36 From Gauss to Gröbner Bases p. 2 Representation, example Consider f 1 = x 4 + x 3 11x 2 9x+ 18 f 2 = x 2 y 3x 2 xy+ 3x 6y+ 18 S ( f 1, f 2 ) = 2x 3 y 5x 2 y 9xy+ 18y+ 3x 4 3x 3 18x 2

37 From Gauss to Gröbner Bases p. 2 Representation, example Consider f 1 = x 4 + x 3 11x 2 9x+ 18 f 2 = x 2 y 3x 2 xy+ 3x 6y+ 18 S ( f 1, f 2 ) = 3x 2 y+ 3xy+ 18y+ 3x 4 3x 3 24x 2 36x +2x f 2

38 From Gauss to Gröbner Bases p. 2 Representation, example Consider f 1 = x 4 + x 3 11x 2 9x+ 18 f 2 = x 2 y 3x 2 xy+ 3x 6y+ 18 S ( f 1, f 2 ) = 3x 4 + 3x 3 33x 2 27x+ 54 +(2x 3) f 2

39 From Gauss to Gröbner Bases p. 2 Representation, example Consider f 1 = x 4 + x 3 11x 2 9x+ 18 f 2 = x 2 y 3x 2 xy+ 3x 6y+ 18 S ( f 1, f 2 ) = 3 f 1 +(2x 3) f 2 lt (3) lt ( f 1 )=3x 4 x 4 y lt (2x 4) lt ( f 2 )=2x 3 y x 4 y h 3 = 0

40 From Gauss to Gröbner Bases p. 2 Nice form? Difficulty: f 1 = xy+ z f 2 = y 2 z =tdeg(x y z) S ( f 1, f 2 )= y z x z S ( f 1, f 2 ) new leading terms Linear analogue: w +2x 5y +z = 0 x +3y +z = 0 z = 0 2w +4x = 0 w +2x 5y +z = 0 x +3y +z = 0 z = 0 10y 2z = 0

41 From Gauss to Gröbner Bases p. 2 Nice form: definition F is a Gröbner basis iff p f 1, f 2,..., f m lt fk lt ( p) for some k= 1,..., m where f 1, f 2,..., f m is the ideal of F in x 1, x 2,..., x n ; that is, h1 f 1 + h 2 f h m f m : h 1, h 2,..., h m x 1, x 2,..., x n

42 From Gauss to Gröbner Bases p. 3 Nice form? result Theorem (Buchberger, 1965) F is a Gröbner basis every S fi, f j has a representation modulo F

43 From Gauss to Gröbner Bases p. 3 Nice form? algorithm Algorithm Gröbner basis (Buchberger, 1965) Input: F= f 1, f 2,..., f m, Output: G= g 1, g 2,..., g n, a Gröbner basis of F wrt 1. G=F 2. Identify critical pairs g i, g j from t i,t j i j 3. Loop through critical pairs: (a) S fi, f j =σi j g i σ j i g j (b) Reduce S gi, g j modulo G, append to G 4. Gröbner basis? YES: Done! NO: go to 2

44 From Gauss to Gröbner Bases p. 3 Part 3: Solutions to questions How do Gröbner bases answer our questions?

45 From Gauss to Gröbner Bases p. 3 Common zeroes? F has common zeroes GBasis(F ) has common zeroes constant GBasis(F ) Linear parallel: x+ y= 0 1=0

46 From Gauss to Gröbner Bases p. 3 Common zeroes? example 1 f 1 = x f 2 = y 2 + 1

47 From Gauss to Gröbner Bases p. 3 Common zeroes? example 1 f 1 = x f 2 = y GBasis(F ) = F S ( f 1, f 2 ) = y 2 x 2 = f 2 f 1 c F for any c COMMON ZEROES

48 From Gauss to Gröbner Bases p. 3 Common zeroes? example 2 f 2 = f 1 + 1

49 From Gauss to Gröbner Bases p. 3 Common zeroes? example 2 f 2 = f S ( f 1, f 2 ) = 1 GBasis(F)={ f 1, f 2, 1} NO COMMON ZEROES

50 From Gauss to Gröbner Bases p. 3 Finitely many solutions? Finitely many zeroes (Zero-dimensional) a power of each variable appears in LeadingTerms(GBasis(F )) Linear parallel: all variables in diagonal x y +z 3 3y +z 5 z = 1

51 From Gauss to Gröbner Bases p. 3 Dimension of zeroes? example 1 f 1 = x f 2 = y x i in f 1, y j in f 2 ZERO-DIMENSIONAL!

52 From Gauss to Gröbner Bases p. 3 Dimension of zeroes? Dimension of Zeroes(F ) = Dimension of Zeroes(GBasis(F )) = Dimension of Zeroes(LeadingTerms(GBasis(F ))

53 From Gauss to Gröbner Bases p. 3 Dimension of zeroes? example 2 f 1 = w x+ f 2 = wy+ f 3 = w z+ x i not a leading term= not zero-dimensional Dimension of Zeroes(F ) = Dimension of Zeroes(w x, w y, w z) = 3 (0, s, t, u) s, t, u

54 From Gauss to Gröbner Bases p. 4 Number of zeroes? Plot lt s on axis count terms not divisible by lt s (Why? Span x i : x i t 1,..., t m = x 1,..., x n f 1, f 2,..., f m )

55 From Gauss to Gröbner Bases p. 4 Number of zeroes? example 1 f 1 = x f 2 = y y 2 x 2

56 From Gauss to Gröbner Bases p. 4 Summary of method f 1 = xy 2 2x y f 2 = x 2 y x 2 + y 1 =lex(y x) S ( f 1, f 2 )= f 1 + f 2 + x 2 2y Add f 3 = x 2 + 2y 2 3

57 From Gauss to Gröbner Bases p. 4 Summary of method f 1 = xy 2 2x y f 2 = x 2 y x 2 + y 1 f 3 = x 2 + 2y 2 3 S ( f 1, f 3 )= f 1 2 f 3 + 2y 4 6y Add f 4 = y 4 3y 2 + 2

58 From Gauss to Gröbner Bases p. 4 Summary of method f 1 = xy 2 2x y f 2 = x 2 y x 2 + y 1 f 3 = x 2 + 2y 2 3 f 4 = y 4 3y S ( f 2, f 3 )= f 3 + 2y 3 2y 2 4y+ 4 Add f 5 = y 3 y 2 2y+ 2

59 From Gauss to Gröbner Bases p. 4 Summary of method f 1 = xy 2 2x y f 2 = x 2 y x 2 + y 1 f 3 = x 2 + 2y 2 3 f 4 = y 4 3y f 5 = y 3 y 2 2y+ 2 GB({ f 1, f 2 })={ f 1, f 2, f 3, f 4, f 5 }

60 From Gauss to Gröbner Bases p. 4 Summary of method leading terms: xy 2, x 2 y, x 2, y 4, y 3 five solutions (in fact f 1 =(x 1) y 2 1 and f 2 = x (y 1))

61 From Gauss to Gröbner Bases p. 4 Part 3: Recent advances Optimizations of Buchberger s algorithm?

62 From Gauss to Gröbner Bases p. 4 Difficulty with Gröbner bases Worst-case: Doubly-exponential in variables, total degree Can we skip some S-polys?

63 From Gauss to Gröbner Bases p. 4 Skip? previous work 1965 Buchberger, B Buchberger, B Gebauer, R. and Möller, H Caboara, M.; Kreuzer, M.; Robbiano, L Faugère, J.C. (criterion on other terms, coefficients)

64 From Gauss to Gröbner Bases p. 4 Skip? example 1 f 1 = x 2 + R 1 f 2 = y 2 + R 2 S ( f 1, f 2 ) = R 1 f 2 R 2 f 2 We can skip S ( f 1, f 2 )!

65 From Gauss to Gröbner Bases p. 4 Skip? first result! Theorem BCRP (Buchberger, 1965) If t 1, t 2 are relatively prime Then can skip S ( f 1, f 2 ) f 1, f 2 with leading terms t 1, t 2 Linear parallel: two equations already in nice form x +2y + 0 3y +

66 From Gauss to Gröbner Bases p. 4 Skip? summary example leading terms: xy 2, x 2 y, x 2, y 4, y 3 S ( f 3, f 4 ) = y 3 S ( f 2, f 3 ) + S ( f 2, f 4 ) S ( f 3, f 5 ) = y 2 S ( f 2, f 3 ) + S ( f 2, f 5 ) We can skip S ( f 3, f 4 ) and S ( f 3, f 5 )!

67 From Gauss to Gröbner Bases p. 4 Skip? second result! Theorem BCLD (Buchberger, 1979) If t 2 divides lcm(t 1, t 3 ) Then can skip S ( f 1, f 3 ) f 1, f 2, f 3 with leading terms t 1, t 2, t 3

68 total-degree term ordering From Gauss to Gröbner Bases p. 5 How common are BC skips? BCRP, BCLD skips not skipped Polys in GB compute GBasis of polynomials f 1, f 2, f 3 two to five variables two to five monomials, total degree 5, exponents from 0 to 10 random coefficients in 2

69 From Gauss to Gröbner Bases p. 5 Skip? example 3 f 1 = w x+ f 2 = wy+ f 3 = w z+ S ( f 1, f 2 ) = 1 f f 2 S ( f 2, f 3 ) = 2 f f 3 S ( f 1, f 3 ) = 2 f 1 3 f 3 Curious pattern...

70 From Gauss to Gröbner Bases p. 5 Skip? Third result! Theorem EBC3 (To appear 2007) If (EC-div) and (EC-var) Then can skip S ( f 1, f 3 ) modulo f 1,f 2,f 3 f 1, f 2, f 3 with leading terms t 1, t 2, t 3

71 From Gauss to Gröbner Bases p. 5 Skip? Third result! Theorem EBC3 (To appear 2007) If (EC-div) and (EC-var) Then can skip S ( f 1, f 3 ) modulo f 1,f 2,f 3 f 1, f 2, f 3 with leading terms t 1, t 2, t 3 (EC-div): gcd(t 1, t 3 ) t 2, or t 2 lcm(t 1, t 3 ) (EC-var): x deg x t 1 = 0, or deg x t 3 = 0, or deg x t 2 max deg x t 1,deg x t 3

72 From Gauss to Gröbner Bases p. 5 How often can we skip? Chained Polynomial Skips vars BCRP, BCLD skips EC skips 3 ~53,000 ~7,000 4 ~43,000 ~6,000 5 ~35,000 ~5,000 6 ~28,000 ~4, ,000 triplets t 1, t 2, t 3 maximum degree of each indeterminate: 10 ordered from least to greatest consistent in lexicographic, total-degree term orderings

73 From Gauss to Gröbner Bases p. 5 Skip? new result! Theorem NCm (Discovered Dec. 2006) If (BCLM) or (EC) Then can skip S f1, f m modulo f1,f 2,...,f m f 1, f 2,..., f m with leading terms t 1, t 2,..., t m

74 From Gauss to Gröbner Bases p. 5 Skip? new result! Theorem NCm (Discovered Dec. 2006) If (BCLM) or (EC) Then can skip S f1, f m modulo f1,f 2,...,f m f 1, f 2,..., f m with leading terms t 1, t 2,..., t m (EC): k : 1 k m gcd t 1, t m tk, and x deg x t 1 = 0, or deg x t 3 = 0, or deg x t 1 deg x t 2 deg x t m, or deg x t 1 deg x t 2 deg x t m

75 From Gauss to Gröbner Bases p. 5 Skip? fifth result! Corollary If t k = x 0 x k for k= 1,..., m and for i= 1,..., m 1 S fi, f i+1 has a representation modulo f1,..., f m, Then f 1,..., f m is GB f 1,..., f m with leading terms t 1,..., t m

76 From Gauss to Gröbner Bases p. 5 Summary nice form? elimination common zeroes? dimension of zeroes? number of zeroes? Gauss elimination does not introduce lt lcm of coefficients, subtract consistent systems variables that are not pivots zero, one, infinite

77 From Gauss to Gröbner Bases p. 5 Summary nice form? elimination common zeroes? dimension of zeroes? number of zeroes? Gröbner S-poly does not introduce lt lcm of lts, subtract no constant in basis dimension of lts terms not divided by lts

78 From Gauss to Gröbner Bases p. 5 Questions 1. GB detection? (Exists such that already GB?) 2. GBs under composition? ( Chain Rule for GB computation?) 3. Criterion in noncommutative rings? (xy y x, with possible application to cryptography)

79 From Gauss to Gröbner Bases p. 5 Finis Thank you!