On the construction of several multivariate resultant matrices - PowerPoint PPT Presentation

On the construction of several multivariate resultant matrices Weikun Sun School of Science Tianjin University of Technology and Education Jan. 1st, 2014

Table of Content 1 Backgrounds and Historical Review 2 Sylvester-Macaulay Type 3 Cayley-Dixon Type 4 Mixed Type

What is the resultant?

What is the resultant? In Gelfand, Kapranov and Zelevinsky’s book (1994) Definition The resultant of k +1 polynomials f 0 , . . . , f k in k variables is de- fined as an irreducible polynomial in the coefficients of f 0 , . . . , f k , which vanishes whenever these polynomials have a common root.

The Problem So the problems here is

The Problem So the problems here is 1 Does the resultant exist?

The Problem So the problems here is 1 Does the resultant exist? 2 If does, how to find it?

The Problem So the problems here is 1 Does the resultant exist? 2 If does, how to find it? The first question was proved in GKZ[1994], and in this talk, we will focus on Problem 2 and give some introductory results.

Main Idea Given a polynomial system F § we want to construct a new poly- nomial system F ′ and use 1 the determinant of F ′ s coefficient matrix; OR 2 the maximal minor of F ′ s coefficient matrix; OR 3 the quotient of maximal minor and extraneous factor to find the resultant of original polynomial system.

Historical Review 1 Sylvester Type Macaulay F. Macaulay[1902], J. Canny[1990] Newton sparse M. Kapranov, B. Sturmfels, A. Zelevinsky[1992] J. Canny, I. Emiris[1994, 1995, 2000] Dixon dialytic A. Chtcherba and D. Kapur[2002]

Historical Review 1 Sylvester Type Macaulay F. Macaulay[1902], J. Canny[1990] Newton sparse M. Kapranov, B. Sturmfels, A. Zelevinsky[1992] J. Canny, I. Emiris[1994, 1995, 2000] Dixon dialytic A. Chtcherba and D. Kapur[2002] 2 Cayley Type Bezout Dixon A. Dixon[1908] D. Kapur, T. Saxena, L. Yang[1994]

Historical Review 1 Sylvester Type Macaulay F. Macaulay[1902], J. Canny[1990] Newton sparse M. Kapranov, B. Sturmfels, A. Zelevinsky[1992] J. Canny, I. Emiris[1994, 1995, 2000] Dixon dialytic A. Chtcherba and D. Kapur[2002] 2 Cayley Type Bezout Dixon A. Dixon[1908] D. Kapur, T. Saxena, L. Yang[1994] 3 Mixed Type A. Dixon[1908] M. Zhang, E. Chionh and R. Goldman[1998] A. Khetan[2003], Sun and Li[2006]

The System Consider the following generic n -degree ( k 1 , k 2 , . . . , k n ) polyno- mial system k 1 k n � � c 0 ,i 1 , ··· ,i n x k 1 1 · · · x k n f 0 ( x 1 , x 2 , . . . , x n ) = · · · n i 1 =0 i n =0 k 1 k n � � c 1 ,i 1 , ··· ,i n x k 1 1 · · · x k n f 1 ( x 1 , x 2 , . . . , x n ) = · · · n i 1 =0 i n =0 . . . k 1 k n � � c n,i 1 , ··· ,i n x k 1 1 · · · x k n f n ( x 1 , x 2 , . . . , x n ) = · · · n i 1 =0 i n =0 where c j,i 1 , ··· ,i n are unrelated parameters.

Sylvester resultant matrix Consider the following ( n + 1)! � n i =1 k i polynomials x σ 1 1 x σ 2 2 · · · x σ n n · [ f 0 f 1 . . . f n ] (1) where = 0 , 1 , . . . , nk 1 − 1; σ 1 σ 2 = 0 , 1 , . . . , k 2 − 1; = 0 , 1 , . . . , 2 k 3 − 1; σ 3 . . . = 0 , 1 , . . . , ( n − 1) k n − 1 σ n Equation (1) represents multiply the original n + 1 polynomials by n ! � n i =1 k i monomials. In these polynomals, the highest degrees of variable x 1 , x 2 , . . . , x n are ( n + 1) k 1 − 1 , 2 k 2 − 1 , . . . , nk n − 1 respectively.

Sylvester resultant matrix So we have ( n + 1)! � n i =1 k i polynomials, each of which consists of ( n + 1)! � n i =1 k i monomials.

Sylvester resultant matrix Let L ( x 1 , x 2 , . . . , x n ) = [ f 0 f n ], then (1) can be ex- f 1 . . . pressed by matrix form as x ( n − 1) k n − 1 · · · x ( n − 1) k n − 1 x nk 1 − 1 [ L x n L · · · L · · · L ] n n 1 T   1 x n     . .   .   = S   x nk n − 1  n .    .   .   x ( n +1) k 1 − 1 · · · x nk n − 1 1 n where the coefficient matrix S is called Sylvester resultant matrix , whose size is ( n + 1)! � n i =1 k i × ( n + 1)! � n i =1 k i , and its determinant det S is called Sylvester resultant.

Cayley resultant matrix Based on the Cayley quotient of { f 0 , f 1 , . . . , f n } � � f 0 ( x 1 , x 2 , . . . , x n ) f 1 ( x 1 , x 2 , . . . , x n ) · · · f n ( x 1 , x 2 , . . . , x n ) � � � � f 0 (¯ x 1 , x 2 , . . . , x n ) f 1 (¯ x 1 , x 2 , . . . , x n ) · · · f n (¯ x 1 , x 2 , . . . , x n ) � � � � · · · · · · · · · · · · � � � � f 0 (¯ x 1 , ¯ x 2 , . . . , ¯ x n ) f 1 (¯ x 1 , ¯ x 2 , . . . , ¯ x n ) · · · f n (¯ x 1 , ¯ x 2 , . . . , ¯ x n ) � � ( x 1 − ¯ x 1 )( x 2 − ¯ x 2 ) · · · ( x n − ¯ x n ) Cayley quotient is actually not a quotient, but a polynomial in two groups of variables x 1 , x 2 , . . . , x n and ¯ x 1 , ¯ x 2 , . . . , ¯ x n .

Cayley resultant matrix In the expression of Cayley quotient, the highest degrees of vari- ables x 1 , x 2 , . . . , x n , ¯ x 1 , ¯ x 2 , . . . , ¯ x n are k 1 − 1 , 2 k 2 − 1 , . . . , nk n − 1 , nk 1 − 1 , ( n − 1) k 2 − 1 , . . . , k n − 1 respectively.

Cayley resultant matrix So in matrix form, it can be expressed as   T   1 1 x n x n ¯         x 2 x 2 ¯     C (2) n n     . .  .   .  . .     x k 1 − 1 x nk 1 − 1 · · · x nk n − 1 x k n − 1 ¯ · · · ¯ 1 n 1 n Here the coefficient matrix is called Cayley resultant matrix , whose size is n ! � n i =1 k i × n ! � n i =1 k i , and its determinant det C is called Cayley resultant.

Comparison Cayley res. matrix · · · Sylvester res. matrix n ! � n ( n + 1)! � n Size i =1 k i · · · i =1 k i Degree n + 1 · · · 1

Comparison Cayley res. matrix · · · Sylvester res. matrix n ! � n ( n + 1)! � n Size i =1 k i · · · i =1 k i Degree n + 1 · · · 1 ⇑ Question: What happens here?

Our Goal The next step is to construct a resultant matrix 1 whose size lies between the size of Cayley and Sylvester re- sultant matrix: n n ( n + 1)! k i × ( n + 1)! � � k i m m i =1 i =1 2 the degree of it’s entry in coefficients of original polynomial system is m (1 < m < n + 1) .

Mixed Cayley-Sylvester resultant matrix The Mixed Cayley-Sylvester resultant matrix employs two key steps in previous procedure.

Cayley quotient step Firstly, we consider following Cayley quotient Φ m � � f n − m +1 ( x 1 , . . . , x n − m +2 , . . . , x n ) · · · f n ( x 1 , . . . , x n − m +2 , . . . , x n ) � � � � f n − m +1 ( x 1 , . . . , x n − m +2 , . . . , ¯ x n ) · · · f n ( x 1 , . . . , x n − m +2 , . . . , ¯ x n ) � � � � . . . � . . . � . . . � � � � f n − m +1 ( x 1 , . . . , ¯ x n − m +2 , . . . , ¯ x n ) · · · f n ( x 1 , . . . , ¯ x n − m +2 , . . . , ¯ x n ) � � m × m ( x n − m +2 − ¯ x n − m +2 ) · · · ( x n − ¯ x n ) So this is a polynomial in variables x 1 , x 2 , . . . , x n − m +1 , x n − m +2 , . . . , x n with degrees mk 1 , mk 2 , . . . , mk n − m +1 , ( m − 1) k n − m +2 − 1 , . . . , k n − 1 and in variables x n − m +2 , . . . , ¯ ¯ x n with degrees k n − m +2 − 1 , . . . , ( m − 1) k n − 1 .

Cayley quotient step ǫ n − m +2 Consider the coefficients of ¯ n − m +2 · · · ¯ x ǫ n n , where x ǫ n − m +2 = 0 , . . . , k n − m +2 − 1 . . . ǫ n = 0 , . . . , ( m − 1) k n − 1 we could have ( m − 1)! � n i = n − m +2 k i polynomials φ ǫ such that k n − m +2 − 1 ( m − 1) k n − 1 � � ǫ n − m +2 x ǫ n Φ m = · · · φ ǫ ( x 1 , x 2 , . . . , x n )¯ x n − m +2 · · · ¯ n ǫ n − m +2 =0 ǫ n =0

Sylvester step Multiply φ ǫ ( x 1 , x 2 , . . . , x n ) by ( n − m + 1)! � n − m +1 k i monomials i =1 ( n − m +1) k n − m +1 − 1 1 , x 1 , x 2 , . . . x n − m +1 , . . . . . . , x k 1 − 1 · · · x n − m +1 1 we could get ( n − m + 1)!( m − 1)! � n i =1 k i polynomials which � n consist of ( n +1)! i =1 k i monomials m ǫ n − m +1 x ǫ 1 1 · · · x n − m +1 φ ǫ ( x 1 , x 2 , . . . , x n ) where 0 ≤ ǫ 1 ≤ k 1 − 1 . . . 0 ≤ ǫ n − m +1 ≤ ( n − m + 1) k n − m +1

Sylvester step Since the Cayley quotient Φ m only uses m polynomials from the original n +1 polynomials, we can repeat this procedure by C m n +1 times. Altogether we can get n +1 ( n − m + 1)!( m − 1)! � n C m i =1 k i n ( n + 1)! � = ( n − m + 1)! m !( n − m + 1)!( m − 1)! k i i =1 n ( n + 1)! � = k i m i =1 � n n is ( n +1)! polynomials, and the number of monomials x i 1 1 · · · x i n i =1 k i . m Rewrite these polynomials in matrix form, the coefficient matrix � n � n is what we want, and its size is ( n +1)! i =1 k i × ( n +1)! i =1 k i . m m