Bilinear systems with two supports: Koszul resultant matrices, - PowerPoint PPT Presentation

Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors July 17, 2018 ıas R. Bender 1 , Jean-Charles Faug` ere 1 , Mat´ Angelos Mantzaflaris 2 & Elias Tsigaridas 1 1 Sorbonne Universit´ e, CNRS , INRIA , Laboratoire d’Informatique de Paris 6, LIP6 , ´ Equipe PolSys , 4 place Jussieu, F-75005, Paris, France 2 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria

Solving 2-bilinear systems Objective Solve (symbolically) square mixed sparse bilinear systems where f 1 , . . . , f r ∈ K [ X , Y ], bilinear in the blocks X and Y , and f r +1 , . . . , f n ∈ K [ X , Z ], bilinear in the blocks X and Z . Take into the account the sparseness Polynomial time wrt the number of solutions Results Koszul-like determinantal formula for the resultant Extension of the Eigenvalue criteria Extension of the Eigenvector criteria 1/13

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n )

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n ) Add f 0 Resultant of ( f 0 , f 1 , . . . , f n )

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n ) Add f 0 Resultant of ( f 0 , f 1 , . . . , f n ) Weyman complex Determinantal formula

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n ) Add f 0 Resultant of ( f 0 , f 1 , . . . , f n ) Weyman complex f 0 ( α ) eigenvalue Determinantal of � formula M 2 , 2 Schur complement � � � M 1 , 1 M 1 , 2 � M 1 , 1 M 1 , 2 → � M 2 , 1 M 2 , 2 0 M 2 , 2

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n ) Add f 0 Resultant of ( f 0 , f 1 , . . . , f n ) Weyman complex f 0 ( α ) eigenvalue Coordinates of α from Determinantal eigenvector of � of � M 2 , 2 formula M 2 , 2 Schur complement � � � M 1 , 1 M 1 , 2 � M 1 , 1 M 1 , 2 → � M 2 , 1 M 2 , 2 0 M 2 , 2 2/13

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n ) Add f 0 Resultant of ( f 0 , f 1 , . . . , f n ) Weyman complex Determinantal formula 2/13

Determinantal formulas and Weyman complex Approach Add a trilinear f 0 ∈ K [ X , Y , Z ] → Sparse resultant of ( f 0 , f 1 , . . . , f n ). The resultant of ( f 0 , f 1 , . . . , f n ) vanishes ⇐ ⇒ the system has a solution over P n x × P n y × P n z . 3/13

Determinantal formulas and Weyman complex Approach Add a trilinear f 0 ∈ K [ X , Y , Z ] → Sparse resultant of ( f 0 , f 1 , . . . , f n ). The resultant of ( f 0 , f 1 , . . . , f n ) vanishes ⇐ ⇒ the system has a solution over P n x × P n y × P n z . Determinantal formula → Resultant = determinant of a matrix. 3/13

Determinantal formulas and Weyman complex Approach Add a trilinear f 0 ∈ K [ X , Y , Z ] → Sparse resultant of ( f 0 , f 1 , . . . , f n ). The resultant of ( f 0 , f 1 , . . . , f n ) vanishes ⇐ ⇒ the system has a solution over P n x × P n y × P n z . Determinantal formula → Resultant = determinant of a matrix. Several works in this direction: (Non-exhaustive!) [Sturmfels, Zelevinsky, 1994], [Canny, Emiris, 1995] [Kapur, Saxena, 1997], [Chtcherba, Kapur, 2000], [D´Andrea, Dickenstein, 2001] 3/13

Determinantal formulas and Weyman complex Approach Add a trilinear f 0 ∈ K [ X , Y , Z ] → Sparse resultant of ( f 0 , f 1 , . . . , f n ). The resultant of ( f 0 , f 1 , . . . , f n ) vanishes ⇐ ⇒ the system has a solution over P n x × P n y × P n z . Determinantal formula → Resultant = determinant of a matrix. Several works in this direction: (Non-exhaustive!) [Sturmfels, Zelevinsky, 1994], [Canny, Emiris, 1995] [Kapur, Saxena, 1997], [Chtcherba, Kapur, 2000], [D´Andrea, Dickenstein, 2001] [Weyman, Zelevinsky, 1994] → determinantal formulas for unmixed multihomogeneous systems using Weyman complexes . [Dickenstein, Emiris, 2003], [Emiris, Mantzaflaris, 2012],[Emiris, Mantzaflaris, Tsigaridas, 2016], [Bus´ e, Mantzaflaris, Tsigaridas, 2017] 3/13

Determinantal formulas and Weyman complex Approach Add a trilinear f 0 ∈ K [ X , Y , Z ] → Sparse resultant of ( f 0 , f 1 , . . . , f n ). Weyman complex → Complex associated to an overdetermined system, parameterized by a vector m . δ n +1 ( m ) δ 1 ( m ) δ 0 ( m ) K • ( m ) : 0 → K n +1 ( m ) → · · · → K 1 ( m ) → K 0 ( m ) → · · · → K − n ( m ) → 0 − − − − − − − − − − − Determinant of K • ( m ) = 0 ⇐ ⇒ the system has a solution → Resultant of the system. 3/13

Determinantal formulas and Weyman complex Approach Add a trilinear f 0 ∈ K [ X , Y , Z ] → Sparse resultant of ( f 0 , f 1 , . . . , f n ). Weyman complex → Complex associated to an overdetermined system, parameterized by a vector m . δ n +1 ( m ) δ 1 ( m ) δ 0 ( m ) K • ( m ) : 0 → K n +1 ( m ) → · · · → K 1 ( m ) → K 0 ( m ) → · · · → K − n ( m ) → 0 − − − − − − − − − − − Determinant of K • ( m ) = 0 ⇐ ⇒ the system has a solution → Resultant of the system. If ( ∀ i �∈ { 0 , 1 } ) K i ( m ) = 0, Determinant of K • ( m ) = Determinant of δ 1 ( m ) → Determinantal formula. δ 1 ( m ) K • ( m ) : 0 → · · · → 0 → K 1 ( m ) − − − → K 0 ( m ) → 0 → · · · → 0 3/13

Determinantal formula for the Resultant Results Weyman complex → determinantal formula for the resultant of square 2-bilinear system + trilinear polynomial. Koszul-like matrix: The elements in the matrix are ± the coefficients of the polynomials. Generalization of Sylvester-like matrices, i.e. ( g 0 , . . . , g n ) �→ � n i =0 g i f i These matrices were used in previous works, [Weyman & Zelevinsky, 1994], [Dickenstein & Emiris, 2003], [Emiris & Mantzaflaris, 2012], [Emiris, Mantzaflaris & Tsigaridas, 2016], [Bus´ e, Mantzaflaris & Tsigaridas, 2017] Number of solutions Size of the Koszul-like matrix over P n x × P n y × P n z � r �� n − r � � r �� n − r � r · ( n − r ) − n y · n z + n +1 ( n x + 1) ( r − n y +1)( n − r − n z +1) n y n z n y n z 4/13

Solving 2-bilinear systems Example       � f 1 := 7 x 0 y 0 + − 8 x 0 y 1 + − 1 x 1 y 0 + 2 x 1 y 1 ∈ K [ X , Y ]   f 2 := − 5 x 0 y 0 + 7 x 0 y 1 + − 1 x 1 y 0 + − 1 x 1 y 1    f 3 := − 6 x 0 z 0 + 9 x 0 z 1 + − 1 x 1 z 0 + − 2 x 1 z 1 ∈ K [ X , Z ] 5/13

Solving 2-bilinear systems Example �  f 0 := 3 x 0 y 0 z 0 + − 1 x 0 y 0 z 1 + − 4 x 0 y 1 z 0 + 2 x 0 y 1 z 1   ∈ K [ X , Y , Z ]   + 1 x 1 y 0 z 0 + 2 x 1 y 0 z 1 + 2 x 1 y 1 z 0 + − 2 x 1 y 1 z 1  � f 1 := 7 x 0 y 0 + − 8 x 0 y 1 + − 1 x 1 y 0 + 2 x 1 y 1 ∈ K [ X , Y ]   f 2 := − 5 x 0 y 0 + 7 x 0 y 1 + − 1 x 1 y 0 + − 1 x 1 y 1    f 3 := − 6 x 0 z 0 + 9 x 0 z 1 + − 1 x 1 z 0 + − 2 x 1 z 1 ∈ K [ X , Z ] 5/13

Solving 2-bilinear systems Example �  f 0 := 3 x 0 y 0 z 0 + − 1 x 0 y 0 z 1 + − 4 x 0 y 1 z 0 + 2 x 0 y 1 z 1   ∈ K [ X , Y , Z ]   + 1 x 1 y 0 z 0 + 2 x 1 y 0 z 1 + 2 x 1 y 1 z 0 + − 2 x 1 y 1 z 1  � f 1 := 7 x 0 y 0 + − 8 x 0 y 1 + − 1 x 1 y 0 + 2 x 1 y 1 ∈ K [ X , Y ]   f 2 := − 5 x 0 y 0 + 7 x 0 y 1 + − 1 x 1 y 0 + − 1 x 1 y 1    f 3 := − 6 x 0 z 0 + 9 x 0 z 1 + − 1 x 1 z 0 + − 2 x 1 z 1 ∈ K [ X , Z ]   5 − 7 1 1   7 − 8 − 1 2     − 1 − 1 − 5 7     7 − 1 − 1 − 5     1 − 2 − 7 8 M =     8 − 2 1 − 7     2 9 − 2 − 2 − 1 2     2 − 2 9 − 2 2 − 1     1 − 6 − 1 2 3 − 4 − 4 2 − 6 − 1 1 3 5/13

Overview α solution of Square 2-bilinear the system system ( f 1 , . . . , f n ) ( f 1 , . . . , f n ) Add f 0 Resultant of ( f 0 , f 1 , . . . , f n ) Weyman complex f 0 ( α ) eigenvalue Determinantal of � M 2 , 2 formula Schur complement � � � M 1 , 1 M 1 , 2 � M 1 , 1 M 1 , 2 → � M 2 , 1 M 2 , 2 0 M 2 , 2 6/13

Solving 2-bilinear systems Eigenvalues & Eigenvectors Motivation From Sylvester-like matrices → multiplication map of f 0 over K [ X , Y , Z ] / � f 1 , . . . , f n � . Solve using eigenvalues and eigenvectors. We do not compute the resultant, we use the structure of the matrix. But we do not have a Sylvester-like matrix... 7/13

Solving 2-bilinear systems Eigenvalues - Main theorem Let M be a matrix such that Res ( f 0 , f 1 , . . . , f n ) divides det ( M ). Consider a m monomial of f 0 such that � � M 1 , 1 M 1 , 2 We can reorder M as , M 2 , 1 M 2 , 2 M 1 , 1 is square and invertible. The elements in diagonal of M 2 , 2 = coefficient of m . Then, for each α solutions of ( f 1 , . . . , f n ) s.t. m ( α ) � = 0, � � → f 0 M 2 , 2 − M 2 , 1 · M − 1 m ( α ) eigenvalue of 1 , 1 · M 1 , 2 (Schur complement) 8/13