Matrix Formula of Differential Resultant for First Order Generic - PowerPoint PPT Presentation

Matrix Formula of Differential Resultant for First Order Generic Ordinary Differential Polynomials Zhi-Yong Zhang Academy of Mathematics and Systems Science Chinese Academy of Sciences Oct, 2012 Joint work with C.M. Yuan and X.S. Gao Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 1 / 26

Outline of the Talk Background Matrix formula with algebraic Macaulay resultant Matrix formula with algebraic sparse resultant Summary Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 2 / 26

Motivation Resultant and Sparse Resultant are basic concepts in algebraic geometry and powerful tools in algebraic elimination theory with important applications. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 3 / 26

Motivation Resultant and Sparse Resultant are basic concepts in algebraic geometry and powerful tools in algebraic elimination theory with important applications. For algebraic case, matrix representation of resultant has stronger forms. A matrix formula of differential resultant is left as an open issue. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 3 / 26

Algebraic Resultant: Sylvester Resultant Given f = a l x l + a l − 1 x l − 1 + · · · + a 1 x + a 0 , and g = b m x m + b m − 1 x m − 1 + · · · + b 1 x + b 0 . � � a l a l − 1 a l − 2 · · · a 0 � � � � a l a l − 1 a l − 2 · · · a 0 � � � ... ... ... ... � � � � � � � a l a l − 1 a l − 2 · · · a 0 � � Res ( f , g ) = � � · · · b m b m − 1 b m − 2 b 0 � � � � b m b m − 1 b m − 2 · · · b 0 � � � � ... ... ... ... � � � � � � b m b m − 1 b m − 2 · · · b 0 � � f ( x ) = 0 , g ( x ) = 0 have a common solution ⇐ ⇒ Res ( f , g ) = 0 Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 4 / 26

Macaulay Multivariate Resultant n + 1 generic polynomials in n variables: F i ( x 1 , . . . , x n ) ( i = 0 , . . . , n ) . Res : a polynomial in the coefficients of F i . For a given system f i ( i = 0 , . . . , n ) , Res ( f 0 , . . . , f n ) = 0 ⇐ ⇒ f i = 0 have a common solution. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 5 / 26

Works on Algebraic Sparse Resultant Gelfand, Kapranov, and Zelevinsky (1991, 1994) introduced the sparse resultant. Sturmfels (1993, 1994) proved basic properties for the sparse resultant. Canny and Emiris (1993, 1995, 2000) gave matrix formulas for sparse resultants and proposed efficient algorithms. D’Andrea (2002) proved the sparse resultant is a quotient of two determinants. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 6 / 26

Works on Differential Resultant Differential Polynomials Ritt (1932): Differential resultant for two univariate diff polynomials. Ferro (1997): Diff-Res as Macaulay resultant. Not complete . Yang-Zeng-Zhang (2009): Diff-Res with Dixon resultant. Rueda-Sendra (2010): Differential resultant for a linear system. Gao-Li-Yuan (2010): Multivariate differential resultants. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 7 / 26

Notations Notations δ : a derivation operator, y n = δ n y . K : ordinary differential field of char. zero K { y } := K [ δ n y ]( n ∈ N ) : diff. poly. ring B j i : All monomials of total degree less than or equal to j with the basis of the first i elements of B = { 1 , y , y 1 , y 2 } . Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 8 / 26

Notations Notations δ : a derivation operator, y n = δ n y . K : ordinary differential field of char. zero K { y } := K [ δ n y ]( n ∈ N ) : diff. poly. ring B j i : All monomials of total degree less than or equal to j with the basis of the first i elements of B = { 1 , y , y 1 , y 2 } . For example, B 2 2 = { 1 , y , y 2 } and B 2 3 = { 1 , y , y 1 , y 2 , yy 1 , y 2 1 } Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 8 / 26

Differential Resultant by Carr` a Ferro Generic Differential Polynomial : P = � M ∈ m s , r u M M • m s , r : differential monomials in Y of order ≤ s and degree ≤ r • u M : differential indeterminates Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 9 / 26

Differential Resultant by Carr` a Ferro Generic Differential Polynomial : P = � M ∈ m s , r u M M • m s , r : differential monomials in Y of order ≤ s and degree ≤ r • u M : differential indeterminates Differential resultant by Carr` a Ferro : For g 1 , g 2 with ord ( g 1 ) = m , ord ( g 2 ) = n , δ Res ( g 1 , g 2 ) is the algebraic Macaulay’s resultant of the system P ( g 1 , g 2 ) = { δ n g 1 , δ n − 1 g 1 , . . . , g 1 , δ m g 2 , δ m − 1 g 2 , . . . , g 2 } in the polynomial ring S m + n = K [ y , δ y , . . . , δ m + n y ] in m + n + 1 variables. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 9 / 26

An example An example : Consider the polynomial system 1 + a 1 y 1 y + a 2 y 2 + a 3 y 1 + a 4 y + a 5 , g 1 = a 0 y 2 1 + b 1 y 1 y + b 2 y 2 + b 3 y 1 + b 4 y + b 5 , g 2 = b 0 y 2 where a i , b i with i = 0 , . . . , 5 are differential indeterminates. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 10 / 26

An example Based on Carr` a Ferro’s definition, the matrix is y 5 y 4 y 3 2 y 1 . . . . . . y 1 2 2  0 d 1 a 0 . . . δ a 5 . . . 0 0  � B 3 . . . . . . 3 δ g 1    0 0 0 δ a 4 δ a 5  . . . . . .    0 0 a 0 0 0  . . . . . . �     B 3 . . . . . . 3 g 1     0 0 0 a 4 a 5 . . . . . .     0 d 2 b 0 δ b 5 0 0 . . . . . . �   B 3   3 δ g 2 . . . . . .     0 0 . . . 0 . . . δ b 4 δ b 5     � 0 0 . . . b 0 . . . 0 0   B 3 3 g 2   . . . . . .   0 0 . . . 0 . . . b 4 b 5 Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 11 / 26

An example Based on Carr` a Ferro’s definition, the matrix is y 5 y 4 y 3 2 y 1 . . . . . . y 1 2 2  0 d 1 a 0 . . . δ a 5 . . . 0 0  � B 3 . . . . . . 3 δ g 1    0 0 0 δ a 4 δ a 5  . . . . . .    0 0 a 0 0 0  . . . . . . �     B 3 . . . . . . 3 g 1     0 0 0 a 4 a 5 . . . . . .     0 d 2 b 0 δ b 5 0 0 . . . . . . �   B 3   3 δ g 2 . . . . . .     0 0 . . . 0 . . . δ b 4 δ b 5     � 0 0 . . . b 0 . . . 0 0   B 3 3 g 2   . . . . . .   0 0 . . . 0 . . . b 4 b 5 Generally, the diff. res. defined by Carr` a Ferro’s is zero. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 11 / 26

Main Contribution We focus on the following generic differential polynomials 1 y d 1 y y d 1 − 1 f 1 = a y 1 + a y y + · · · + a 0 , d 1 d 1 − 1 1 1 1 y d 2 y y d 2 − 1 f 2 = b y 1 + b y y + · · · + b 0 , d 2 d 2 − 1 1 1 where 1 ≤ d 1 ≤ d 2 , a y 1 , . . . , a 0 , b y 1 , . . . , b 0 are diff. indeterminates. d 1 d 2 Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 12 / 26

Main Contribution We focus on the following generic differential polynomials 1 y d 1 y y d 1 − 1 f 1 = a y 1 + a y y + · · · + a 0 , d 1 d 1 − 1 1 1 1 y d 2 y y d 2 − 1 f 2 = b y 1 + b y y + · · · + b 0 , d 2 d 2 − 1 1 1 where 1 ≤ d 1 ≤ d 2 , a y 1 , . . . , a 0 , b y 1 , . . . , b 0 are diff. indeterminates. d 1 d 2 The methods: Algebraic Macaulay Resultant Algebraic Sparse Resultant ց ւ Matrix formula of Differential Resultant Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 12 / 26

Matrix formula with algebraic Macaulay resultant Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 13 / 26

Matrix formula with algebraic Macaulay resultant Construction of the matrix: 1. Consider the monomial set of the constructed matrix E = B D 3 ∪ y 2 B D − 1 3 with D = 2 d 1 + 2 d 2 − 3. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 14 / 26

Matrix formula with algebraic Macaulay resultant Construction of the matrix: 1. Consider the monomial set of the constructed matrix E = B D 3 ∪ y 2 B D − 1 3 with D = 2 d 1 + 2 d 2 − 3. We will show that if using E as the column monomial set, a nonsingular square matrix can be constructed. Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 14 / 26

Matrix formula with algebraic Macaulay resultant 2. Choosing main monomials Main monomial of p : mm ( p ) . mm ( p 1 ) = y 2 y d 1 − 1 , mm ( p 2 ) = y d 2 1 , mm ( p 3 ) = y d 1 , mm ( p 4 ) = 1 , 1 where p 1 = δ f 1 , p 2 = δ f 2 , p 3 = f 1 , p 4 = f 2 . Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 15 / 26