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A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Kim Batselier KU Leuven Department of Electrical Engineering

  1. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Building blocks of multivariate polynomials? Monomials! 1 , x 1 , x 2 , x 3 , x 2 1 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 2 3 , . . . ordering deg( x 2 1 ) = deg( x 2 x 3 ) = 2 Example f 1 = 2 . 76 x 2 1 − 5 . 51 x 1 x 3 13 / 57

  2. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Building blocks of multivariate polynomials? Monomials! 1 , x 1 , x 2 , x 3 , x 2 1 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 2 3 , . . . ordering deg( x 2 1 ) = deg( x 2 x 3 ) = 2 Example f 1 = 2 . 76 x 2 1 − 5 . 51 x 1 x 3 − 1 . 12 x 1 13 / 57

  3. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Building blocks of multivariate polynomials? Monomials! 1 , x 1 , x 2 , x 3 , x 2 1 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 2 3 , . . . ordering deg( x 2 1 ) = deg( x 2 x 3 ) = 2 Example f 1 = 2 . 76 x 2 1 − 5 . 51 x 1 x 3 − 1 . 12 x 1 + 1 . 99 13 / 57

  4. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Building blocks of multivariate polynomials? Monomials! 1 , x 1 , x 2 , x 3 , x 2 1 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 2 3 , . . . ordering deg( x 2 1 ) = deg( x 2 x 3 ) = 2 Example f 1 = 2 . 76 x 2 1 − 5 . 51 x 1 x 3 − 1 . 12 x 1 + 1 . 99 degree of f 1 = deg ( f 1 ) = 2 13 / 57

  5. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Vector Representation Each monomial corresponds with a vector, each orthogonal with respect to all the others: x 2 . . . x 1 1 C n d : vector space of all polynomials in n variables with complex coefficients up to a degree d 14 / 57

  6. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors A blast from the past Y (0,1) X (1,0) 15 / 57

  7. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 16 / 57

  8. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 1 ∼ ( 1 0 0 0 0 . . . ) 16 / 57

  9. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 1 ∼ ( 1 0 0 0 0 . . . ) ∼ ( 0 1 0 0 0 ) x 1 . . . 16 / 57

  10. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 1 ∼ ( 1 0 0 0 0 . . . ) ∼ ( 0 1 0 0 0 ) x 1 . . . x 2 ∼ ( 0 0 1 0 0 . . . ) 16 / 57

  11. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 1 ∼ ( 1 0 0 0 0 . . . ) ∼ ( 0 1 0 0 0 ) x 1 . . . x 2 ∼ ( 0 0 1 0 0 . . . ) ∼ ( 0 0 0 1 0 ) x 3 . . . 16 / 57

  12. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 1 ∼ ( 1 0 0 0 0 . . . ) ∼ ( 0 1 0 0 0 ) x 1 . . . x 2 ∼ ( 0 0 1 0 0 . . . ) ∼ ( 0 0 0 1 0 ) x 3 . . . x 2 ∼ ( 0 0 0 0 1 . . . ) 1 16 / 57

  13. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Each monomial is described by a coefficient vector: 1 ∼ ( 1 0 0 0 0 . . . ) ∼ ( 0 1 0 0 0 ) x 1 . . . x 2 ∼ ( 0 0 1 0 0 . . . ) ∼ ( 0 0 0 1 0 ) x 3 . . . x 2 ∼ ( 0 0 0 0 1 . . . ) 1 . . . 16 / 57

  14. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Coefficient vector of multivariate polynomial 2 . 76 x 2 = 1 − 5 . 51 x 1 x 3 − 1 . 12 x 1 + 1 . 99 f 1 17 / 57

  15. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Coefficient vector of multivariate polynomial 2 . 76 x 2 = 1 − 5 . 51 x 1 x 3 − 1 . 12 x 1 + 1 . 99 f 1 � � ∼ 2 . 76 0 0 0 0 1 0 0 0 0 0 � � − 5 . 51 0 0 0 0 0 0 1 0 0 0 � � − 1 . 12 0 1 0 0 0 0 0 0 0 0 � � +1 . 99 1 0 0 0 0 0 0 0 0 0 17 / 57

  16. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Polynomials as Vectors Coefficient vector of multivariate polynomial 2 . 76 x 2 = 1 − 5 . 51 x 1 x 3 − 1 . 12 x 1 + 1 . 99 f 1 � � ∼ 2 . 76 0 0 0 0 1 0 0 0 0 0 � � − 5 . 51 0 0 0 0 0 0 1 0 0 0 � � − 1 . 12 0 1 0 0 0 0 0 0 0 0 � � +1 . 99 1 0 0 0 0 0 0 0 0 0 � � ∼ 1 . 99 − 1 . 12 0 0 2 . 76 0 − 5 . 51 0 0 0 f 1 17 / 57

  17. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Operations on Polynomials Addition of Polynomials Addition of vectors: f 1 f 2 18 / 57

  18. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Operations on Polynomials Addition of Polynomials Addition of vectors: f 1 f 2 19 / 57

  19. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Operations on Polynomials Addition of Polynomials Addition of vectors: f 1 + f 2 f 1 f 2 20 / 57

  20. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of 2 multivariate polynomials h, f ∈ C n d 21 / 57

  21. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of 2 multivariate polynomials h, f ∈ C n d f × h 21 / 57

  22. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of 2 multivariate polynomials h, f ∈ C n d f × h f × ( h 0 + h 1 x 1 + h 2 x 2 + . . . + h q x d h = n ) 21 / 57

  23. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of 2 multivariate polynomials h, f ∈ C n d f × h f × ( h 0 + h 1 x 1 + h 2 x 2 + . . . + h q x d h = n ) h 0 f + h 1 x 1 f + h 2 x 2 f + . . . + h q x d h = n f 21 / 57

  24. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of 2 multivariate polynomials h, f ∈ C n d f × h f × ( h 0 + h 1 x 1 + h 2 x 2 + . . . + h q x d h = n ) h 0 f + h 1 x 1 f + h 2 x 2 f + . . . + h q x d h = n f   f x 1 f     x 2 f � � ∼ h 0 h 1 h 2 . . . h q    .  .   .   x d h n f 21 / 57

  25. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of 2 multivariate polynomials h, f ∈ C n d f × h f × ( h 0 + h 1 x 1 + h 2 x 2 + . . . + h q x d h = n ) h 0 f + h 1 x 1 f + h 2 x 2 f + . . . + h q x d h = n f   f x 1 f     x 2 f � � ∼ h 0 h 1 h 2 . . . h q    .  .   .   x d h n f ∼ h M f 21 / 57

  26. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication Example f = x 1 x 2 − x 2 and h = x 2 1 + 2 x 2 − 9 .   f   x 1 f      x 2 f    � � h M f = − 9 0 2 1 0 0 .   x 2 1 f       x 1 x 2 f       x 2 2 f 22 / 57

  27. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication Example M f = x 2 x 2 x 3 x 2 x 1 x 2 x 3 x 4 x 3 x 2 1 x 2 x 1 x 3 x 4 1 x 1 x 2 x 1 x 2 1 x 2 1 x 2 1 2 1 2 2 1 2 1 2 0 1 f 0 0 − 1 0 1 0 0 0 0 0 0 0 0 0 0 x 1 f 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 0 B C B C x 2 f 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 B C x 2 1 f B C 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 0 B C x 1 x 2 f @ 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 A x 2 2 f 0 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 23 / 57

  28. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication Example M f = x 2 x 2 x 3 x 2 x 1 x 2 x 3 x 4 x 3 x 2 1 x 2 x 1 x 3 x 4 1 x 1 x 2 x 1 x 2 1 x 2 1 x 2 1 2 1 2 2 1 2 1 2 0 1 f 0 0 − 1 0 1 0 0 0 0 0 0 0 0 0 0 x 1 f 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 0 B C B C x 2 f 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 B C x 2 1 f B C 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 0 B C x 1 x 2 f @ 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 A x 2 2 f 0 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 hM f = x 2 x 2 x 3 x 2 x 1 x 2 x 3 x 4 x 3 x 2 1 x 2 x 1 x 3 x 4 “ 1 x 1 x 2 x 1 x 2 1 x 2 1 x 2 1 2 1 2 2 1 2 1 2 ” 0 0 9 0 − 9 − 2 0 − 1 2 0 0 1 0 0 0 23 / 57

  29. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication Example M f = x 2 x 2 x 3 x 2 x 1 x 2 x 3 x 4 x 3 x 2 1 x 2 x 1 x 3 x 4 1 x 1 x 2 x 1 x 2 1 x 2 1 x 2 1 2 1 2 2 1 2 1 2 0 1 f 0 0 − 1 0 1 0 0 0 0 0 0 0 0 0 0 x 1 f 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 0 B C B C x 2 f 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 B C x 2 1 f B C 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 0 B C x 1 x 2 f @ 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 0 A x 2 2 f 0 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0 hM f = x 2 x 2 x 3 x 2 x 1 x 2 x 3 x 4 x 3 x 2 1 x 2 x 1 x 3 x 4 “ 1 x 1 x 2 x 1 x 2 1 x 2 1 x 2 1 2 1 2 2 1 2 1 2 ” 0 0 9 0 − 9 − 2 0 − 1 2 0 0 1 0 0 0 ∼ 9 x 2 − 9 x 1 x 2 − 2 x 2 2 − x 2 1 x 2 + 2 x 1 x 2 2 + x 3 1 x 2 23 / 57

  30. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Multiplication of multivariate polynomials Multiplication of Polynomials Every possible multiplication of f lies in a vector space M f spanned by f, x 1 f, x 2 f, . . . x 1 f .... f x d h n f M f 24 / 57

  31. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Definition multivariate polynomials Fix any monomial order > on C n d and let F = ( f 1 , . . . , f s ) be a s-tuple of polynomials in C n d . Then every p ∈ C n d can be written as p = h 1 f 1 + . . . + h s f s + r where h i , r ∈ C n d . For each i, h i f i = 0 or LM( p ) ≥ LM( h i f i ) , and either r = 0 , or r is a linear combination of monomials, none of which is divisible by any of LM( f 1 ) , . . . , LM( f s ) . 25 / 57

  32. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Definition multivariate polynomials Fix any monomial order > on C n d and let F = ( f 1 , . . . , f s ) be a s-tuple of polynomials in C n d . Then every p ∈ C n d can be written as p = h 1 f 1 + . . . + h s f s + r where h i , r ∈ C n d . For each i, h i f i = 0 or LM( p ) ≥ LM( h i f i ) , and either r = 0 , or r is a linear combination of monomials, none of which is divisible by any of LM( f 1 ) , . . . , LM( f s ) . Differences with division of numbers Remainder r depends on the way we order monomials Dividends h 1 , . . . , h s and remainder r depend on order of divisors f 1 , . . . , f s 25 / 57

  33. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Describing the quotient p = h 1 f 1 + . . . + h s f s + r 26 / 57

  34. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Describing the quotient p = h 1 f 1 + . . . + h s f s + r 26 / 57

  35. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Describing the quotient p = h 1 f 1 + . . . + h s f s + r   f 1 x 1 f 1     x 2 f 1 � � h 10 h 11 h 12 . . . h 1 q    .  .   .   x d 1 n f 1 26 / 57

  36. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Describing the quotient p = h 1 f 1 + . . . + h s f s + r   f k x 1 f k     x 2 f k � � h k 0 h k 1 h k 2 . . . h kw    .  .   .   x d k n f k 27 / 57

  37. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Describing the quotient p = h 1 f 1 + . . . + h s f s + r   f s x 1 f s     x 2 f s � � h s 0 h s 1 h s 2 . . . h sv    .  .   .   x d s n f s 28 / 57

  38. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Describing the quotient p = h 1 f 1 + . . . + h s f s + r   f 1 x 1 f 1     x 2 f 1    .  .   .   � �  x d 1  h 10 h 11 h 12 . . . h 1 q h 20 h 21 . . . h sv n f 1     f 2     x 1 f 2     . .   .   x d s n f s 29 / 57

  39. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Divisor Matrix D Given a set of polynomials f 1 , . . . , f s ∈ C n d , each of degree d i ( i = 1 . . . s ) and a polynomial p ∈ C n d of degree d then the divisor matrix D is given by 0 1 f 1 x 1 f 1 B C B C B C x 2 f 1 B C B C . B C . . B C B C B x d 1 C D = n f 1 B C B C f 2 B C B C B C x 1 f 2 B C B C . B C . . B C @ A x d s n f s where each polynomial f i is multiplied with all monomials x α i from degree 0 up to degree k i = deg( p ) − deg( f i ) such that x α i LM( f i ) ≤ LM( p ) . 30 / 57

  40. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials Example Divisor Matrix To divide p = 4 + 5 x 1 − 3 x 2 − 9 x 2 1 + 7 x 1 x 2 by f 1 = − 2 + x 1 + x 2 , f 2 = 3 − x 1 : x 2 1 x 1 x 2 x 1 x 2 1   f 1 − 2 1 1 0 0 0 − 2 0 1 1 x 1 f 1     D = f 2 3 − 1 0 0 0     0 3 0 − 1 0 x 1 f 2   x 2 f 2 0 0 3 0 − 1 31 / 57

  41. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials D 32 / 57

  42. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials R D 33 / 57

  43. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials R p D 34 / 57

  44. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials Basis Operations in the Framework Division of Multivariate Polynomials R r p h 1 f 1 + . . . + h s f s D 35 / 57

  45. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Outline Introduction 1 Basis Operations in the Framework 2 ”Advanced” Operations in the Framework 3 Conclusions and Future Work 4 36 / 57

  46. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Macaulay Matrix ”Advanced” operations on polynomials Eliminate variables Compute a least common multiple of 2 multivariate polynomials Compute a greatest common divisor of 2 multivariate polynomials One More Key Player: Macaulay matrix 37 / 57

  47. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Macaulay Matrix Macaulay Matrix Given a set of multivariate polynomials f 1 , . . . , f s , each of degree d i ( i = 1 . . . s ) then the Macaulay matrix of degree d is given by   f 1  x 1 f 1    .   .   .     x d − d 1 f 1   n M ( d ) =    f 2      x 1 f 2     .  .  .     x d − d s f s n where each polynomial f i is multiplied with all monomials up to degree d − d i for all i = 1 . . . s . 38 / 57

  48. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Macaulay Matrix Row space of the Macaulay matrix M d = { h 1 f 1 + h 2 f 2 + . . . + h s f s | for all possible h 1 , h 2 , . . . , h s with degrees d − d 1 , d − d 2 , . . . , d − d s respectively } 39 / 57

  49. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Macaulay Matrix Row space of the Macaulay matrix M d = { h 1 f 1 + h 2 f 2 + . . . + h s f s | for all possible h 1 , h 2 , . . . , h s with degrees d − d 1 , d − d 2 , . . . , d − d s respectively } x 1 f 1 f 2 . . . f 1 . . . x d − ds f s n M d 39 / 57

  50. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Macaulay Matrix For the following polynomial system: � f 1 : x 1 x 2 − 2 x 2 = 0 f 2 : x 2 − 3 = 0 the Macaulay matrix of degree 3 is x 2 x 2 x 3 x 2 x 1 x 2 x 3 1 x 1 x 2 x 1 x 2 1 x 2 1 2 1 2 2 0 0 − 2 0 1 0 0 0 0 0 f 1 0 1 x 1 f 1 0 0 0 0 − 2 0 0 1 0 0 B C 0 0 0 0 0 − 2 0 0 1 0 x 2 f 1 B C B C f 2 B − 3 0 1 0 0 0 0 0 0 0 C B C M (3) = 0 − 3 0 0 1 0 0 0 0 0 x 1 f 2 B C B C x 2 f 2 0 0 − 3 0 0 1 0 0 0 0 B C B C x 2 0 0 0 − 3 0 0 0 1 0 0 1 f 2 B C B C x 1 x 2 f 2 0 0 0 0 − 3 0 0 0 1 0 @ A x 2 0 0 0 0 0 − 3 0 0 0 1 2 f 2 40 / 57

  51. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Macaulay Matrix Sparsity pattern M (10) 41 / 57

  52. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Elimination Problem Given a set of multivariate polynomials f 1 , . . . , f s and x e � { x 1 , . . . , x n } . Find a polynomial g = h 1 f 1 + . . . + h s f s that does not contain any of the x e variables. 42 / 57

  53. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Elimination Problem Given a set of multivariate polynomials f 1 , . . . , f s and x e � { x 1 , . . . , x n } . Find a polynomial g = h 1 f 1 + . . . + h s f s that does not contain any of the x e variables. Example From the following polynomial system in 3 variables x 1 , x 2 , x 3 :  x 2 f 1 = 1 + x 2 + x 3 − 1 ,  x 1 + x 2 = 2 + x 3 − 1 , f 2 x 1 + x 2 + x 2 f 3 = 3 − 1 ,  we want to find a g = h 1 f 1 + h 2 f 2 + h 3 f 3 only in x 3 . 42 / 57

  54. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Example Since g = h 1 f 1 + h 2 f 2 + h 3 f 3 , it lies in x 1 f 1 f 2 . . . f 1 . . . x d − ds f s n M d for a certain degree d . 43 / 57

  55. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Example Also, since g only contains the variables x 3 , it is built up from the monomial basis x 2 3 . . . x 3 1 up to a certain degree d . 44 / 57

  56. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Example We will call this vector space that is spanned by the variables x 3 E d : E d 44 / 57

  57. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Example g ∈ M d and g ∈ E d ; hence g lies in the intersection M d ∩ E d : M d g E d for some particular degree d . 45 / 57

  58. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Finding the intersection M d v 2 θ 2 u 2 o E d θ 1 = 0 u 1 = v 1 46 / 57

  59. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Elimination Example We revisit  x 2 1 + x 2 + x 3 = 1 ,  x 1 + x 2 2 + x 3 = 1 , x 1 + x 2 + x 2 = 1 .  3 we eliminate both x 1 and x 2 d = 6 , g ( x 3 ) = x 2 3 − 4 x 3 3 + 4 x 4 3 − x 6 3 . we eliminate x 2 : d = 2 , g ( x 1 , x 3 ) = x 1 − x 3 − x 2 1 + x 2 3 . 47 / 57

  60. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Least Common Multiple Least Common Multiple A multivariate polynomial l is called a least common multiple (LCM) of 2 multivariate polynomials f 1 , f 2 if 1 f 1 divides l and f 2 divides l . 2 l divides any polynomial which both f 1 and f 2 divide. f 1 f 2 48 / 57

  61. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Least Common Multiple Least Common Multiple A multivariate polynomial l is called a least common multiple (LCM) of 2 multivariate polynomials f 1 , f 2 if 1 f 1 divides l and f 2 divides l . 2 l divides any polynomial which both f 1 and f 2 divide. f 1 f 2 l = LCM ( f 1 , f 2 ) 48 / 57

  62. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Least Common Multiple Finding the LCM The LCM l of f 1 and f 2 satisfies: LCM( f 1 , f 2 ) � l = f 1 h 1 = f 2 h 2 49 / 57

  63. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Least Common Multiple Finding the LCM The LCM l of f 1 and f 2 satisfies: LCM( f 1 , f 2 ) � l = f 1 h 1 = f 2 h 2 M f 2 LCM ( f 1 , f 2 ) o M f 1 49 / 57

  64. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Greatest Common Divisor A multivariate polynomial g is called a greatest common divisor of 2 multivariate polynomials f 1 and f 2 if 1 g divides f 1 and f 2 . 2 If p is any polynomial which divides both f 1 and f 2 , then p divides g . f 1 f 2 50 / 57

  65. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Greatest Common Divisor A multivariate polynomial g is called a greatest common divisor of 2 multivariate polynomials f 1 and f 2 if 1 g divides f 1 and f 2 . 2 If p is any polynomial which divides both f 1 and f 2 , then p divides g . f 1 f 2 g = GCD ( f 1 , f 2 ) 50 / 57

  66. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Finding the GCD Remember that LCM( f 1 , f 2 ) � l = f 1 h 1 = f 2 h 2 . We also have that f 1 f 2 = l g, with g � GCD( f 1 , f 2 ) . 51 / 57

  67. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Finding the GCD Remember that LCM( f 1 , f 2 ) � l = f 1 h 1 = f 2 h 2 . We also have that f 1 f 2 = l g, with g � GCD( f 1 , f 2 ) . Answer: g = f 1 f 2 = f 1 = f 2 . l h 2 h 1 51 / 57

  68. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Blind Image Deconvolution F 1 ( z 1 , z 2 ) = I ( z 1 , z 2 ) D 1 ( z 1 , z 2 ) + N 1 ( z 1 , z 2 ) F 2 ( z 1 , z 2 ) = I ( z 1 , z 2 ) D 2 ( z 1 , z 2 ) + N 2 ( z 1 , z 2 ) I ( z 1 , z 2 ) = τ -GCD ( F 1 , F 2 ) F 1 ( z 1 , z 2 ) F 2 ( z 1 , z 2 ) 52 / 57

  69. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Blind Image Deconvolution F 1 ( z 1 , z 2 ) = I ( z 1 , z 2 ) D 1 ( z 1 , z 2 ) + N 1 ( z 1 , z 2 ) F 2 ( z 1 , z 2 ) = I ( z 1 , z 2 ) D 2 ( z 1 , z 2 ) + N 2 ( z 1 , z 2 ) I ( z 1 , z 2 ) = τ -GCD ( F 1 , F 2 ) F 1 ( z 1 , z 2 ) F 2 ( z 1 , z 2 ) τ -GCD ( F 1 , F 2 ) 52 / 57

  70. A Numerical Linear Algebra Framework for Solving Problems with Multivariate Polynomials ”Advanced” Operations in the Framework Greatest Common Divisor Other Operations worked out in the thesis 53 / 57

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