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 f0, . . . , fk in k variables is de- fined as an irreducible polynomial in the coefficients of f0, . . . , fk, 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 (k1, k2, . . . , kn) polyno- mial system

f0(x1, x2, . . . , xn) =

k1

• i1=0

· · ·

kn

• in=0

c0,i1,··· ,inxk1

1 · · · xkn n

f1(x1, x2, . . . , xn) =

k1

• i1=0

· · ·

kn

• in=0

c1,i1,··· ,inxk1

1 · · · xkn n

. . . fn(x1, x2, . . . , xn) =

k1

• i1=0

· · ·

kn

• in=0

cn,i1,··· ,inxk1

1 · · · xkn n

where cj,i1,··· ,in are unrelated parameters.

Sylvester resultant matrix

Consider the following (n + 1)! n

i=1 ki polynomials

xσ1

1 xσ2 2 · · · xσn n · [f0

f1 . . . fn] (1) where σ1 = 0, 1, . . . , nk1 − 1; σ2 = 0, 1, . . . , k2 − 1; σ3 = 0, 1, . . . , 2k3 − 1; . . . σn = 0, 1, . . . , (n − 1)kn − 1 Equation (1) represents multiply the original n + 1 polynomials by n! n

i=1 ki monomials.

In these polynomals, the highest degrees of variable x1, x2, . . . , xn are (n + 1)k1 − 1, 2k2 − 1, . . . , nkn − 1 respectively.

Sylvester resultant matrix

So we have (n + 1)! n

i=1 ki polynomials, each of which consists

• f (n + 1)! n

i=1 ki monomials.

Sylvester resultant matrix

Let L(x1, x2, . . . , xn) = [f0 f1 . . . fn], then (1) can be ex- pressed by matrix form as [L xnL · · · x(n−1)kn−1

n

L · · · xnk1−1

1

· · · x(n−1)kn−1

n

L] =           1 xn . . . xnkn−1

n .

. . x(n+1)k1−1

1

· · · xnkn−1

n

         

T

S where the coefficient matrix S is called Sylvester resultant matrix, whose size is (n + 1)! n

i=1 ki × (n + 1)! n i=1 ki, and its

determinant det S is called Sylvester resultant.

Cayley resultant matrix

Based on the Cayley quotient of {f0, f1, . . . , fn}

• f0(x1, x2, . . . , xn)

f1(x1, x2, . . . , xn) · · · fn(x1, x2, . . . , xn) f0(¯ x1, x2, . . . , xn) f1(¯ x1, x2, . . . , xn) · · · fn(¯ x1, x2, . . . , xn) · · · · · · · · · · · · f0(¯ x1, ¯ x2, . . . , ¯ xn) f1(¯ x1, ¯ x2, . . . , ¯ xn) · · · fn(¯ x1, ¯ x2, . . . , ¯ xn)

• (x1 − ¯

x1)(x2 − ¯ x2) · · · (xn − ¯ xn) Cayley quotient is actually not a quotient, but a polynomial in two groups of variables x1, x2, . . . , xn and ¯ x1, ¯ x2, . . . , ¯ xn.

Cayley resultant matrix

In the expression of Cayley quotient, the highest degrees of vari- ables x1, x2, . . . , xn, ¯ x1, ¯ x2, . . . , ¯ xn are k1 − 1, 2k2 − 1, . . . , nkn − 1, nk1 − 1, (n − 1)k2 − 1, . . . , kn − 1 respectively.

Cayley resultant matrix

So in matrix form, it can be expressed as        1 xn x2

n

. . . xk1−1

1

· · · xnkn−1

n

      

T

C        1 ¯ xn ¯ x2

n

. . . ¯ xnk1−1

1

· · · ¯ xkn−1

n

       (2) Here the coefficient matrix is called Cayley resultant matrix, whose size is n! n

i=1 ki × n! n i=1 ki, and its determinant det C

is called Cayley resultant.

Comparison

Cayley res. matrix · · · Sylvester res. matrix Size n! n

i=1 ki

· · · (n + 1)! n

i=1 ki

Degree n + 1 · · · 1

Comparison

Cayley res. matrix · · · Sylvester res. matrix Size n! n

i=1 ki

· · · (n + 1)! n

i=1 ki

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 + 1)! m

n

• i=1

ki × (n + 1)! m

n

• i=1

ki

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 Φm

• fn−m+1(x1, . . . , xn−m+2, . . . , xn)

· · · fn(x1, . . . , xn−m+2, . . . , xn) fn−m+1(x1, . . . , xn−m+2, . . . , ¯ xn) · · · fn(x1, . . . , xn−m+2, . . . , ¯ xn) . . . . . . . . . fn−m+1(x1, . . . , ¯ xn−m+2, . . . , ¯ xn) · · · fn(x1, . . . , ¯ xn−m+2, . . . , ¯ xn)

• m×m

(xn−m+2 − ¯ xn−m+2) · · · (xn − ¯ xn)

So this is a polynomial in variables x1, x2, . . . , xn−m+1, xn−m+2, . . . , xn with degrees mk1, mk2, . . . , mkn−m+1, (m − 1)kn−m+2 − 1, . . . , kn − 1 and in variables ¯ xn−m+2, . . . , ¯ xn with degrees kn−m+2 − 1, . . . , (m − 1)kn − 1.

Cayley quotient step

Consider the coefficients of ¯ x

ǫn−m+2 n−m+2 · · · ¯

xǫn

n , where

ǫn−m+2 = 0, . . . , kn−m+2 − 1 . . . ǫn = 0, . . . , (m − 1)kn − 1 we could have (m − 1)! n

i=n−m+2 ki polynomials φǫ such that

Φm =

kn−m+2−1

• ǫn−m+2=0

· · ·

(m−1)kn−1

• ǫn=0

φǫ(x1, x2, . . . , xn)¯ x

ǫn−m+2 n−m+2 · · · ¯

xǫn

n

Sylvester step

Multiply φǫ(x1, x2, . . . , xn) by (n − m + 1)! n−m+1

i=1

ki monomials 1, x1, x2, . . . xn−m+1, . . . . . . , xk1−1

1

· · · x

(n−m+1)kn−m+1−1 n−m+1

we could get (n − m + 1)!(m − 1)! n

i=1 ki polynomials which

consist of (n+1)!

m

n

i=1 ki monomials

xǫ1

1 · · · x ǫn−m+1 n−m+1φǫ(x1, x2, . . . , xn)

where 0 ≤ ǫ1 ≤ k1 − 1 . . . 0 ≤ ǫn−m+1 ≤ (n − m + 1)kn−m+1

Sylvester step

Since the Cayley quotient Φm only uses m polynomials from the

• riginal n+1 polynomials, we can repeat this procedure by Cm

n+1

• times. Altogether we can get

Cm

n+1(n − m + 1)!(m − 1)! n i=1 ki

= (n + 1)! (n − m + 1)!m!(n − m + 1)!(m − 1)!

n

• i=1

ki = (n + 1)! m

n

• i=1

ki polynomials, and the number of monomials xi1

1 · · · xin n is (n+1)! m

n

i=1 ki.

Rewrite these polynomials in matrix form, the coefficient matrix is what we want, and its size is (n+1)!

m

n

i=1 ki × (n+1)! m

n

i=1 ki.

Comparison

If m = 1, Φm and φǫ actually are polynomials f0, f1, . . . , fm, and Mixed Cayley-Sylvester resultant matrix coincides with Sylvester resultant matrix. (Up to some interchanges of rows

• r columns)

If m = n + 1, we don’t apply the Sylvester step, and Mixed Cayley-Sylvester resultant matrix coincides with Cayley re- sultant matrix. (Up to some interchanges of rows or column- s) Mixed resultant matrices lie between classical Sylvester and Cayley resultant matrices. 1 2, 3, . . . . . . , n

• n + 1

Sylvester Mixed C-S Cayley

Comparison

If m = 1, Φm and φǫ actually are polynomials f0, f1, . . . , fm, and Mixed Cayley-Sylvester resultant matrix coincides with Sylvester resultant matrix. (Up to some interchanges of rows

• r columns)

If m = n + 1, we don’t apply the Sylvester step, and Mixed Cayley-Sylvester resultant matrix coincides with Cayley re- sultant matrix. (Up to some interchanges of rows or column- s) Mixed resultant matrices lie between classical Sylvester and Cayley resultant matrices. 1 2, 3, . . . . . . , n

• n + 1

Sylvester Mixed C-S Cayley

Comparison

If m = 1, Φm and φǫ actually are polynomials f0, f1, . . . , fm, and Mixed Cayley-Sylvester resultant matrix coincides with Sylvester resultant matrix. (Up to some interchanges of rows

• r columns)

If m = n + 1, we don’t apply the Sylvester step, and Mixed Cayley-Sylvester resultant matrix coincides with Cayley re- sultant matrix. (Up to some interchanges of rows or column- s) Mixed resultant matrices lie between classical Sylvester and Cayley resultant matrices. 1 2, 3, . . . . . . , n

• n + 1

Sylvester Mixed C-S Cayley

Block structure

In our construction, Cayley quotient: combine the coefficients of original polyno- mials; Sylvester step: shift the small block submatrix For Mixed Cayley-Sylvester resultant matrix, it looks like Sylvester resultant matrix in large scale inside it, every block looks like Cayley resultant matrix

Block Structure

Example

For a generic n-degree polynomial system F which contains n+1 polynomials (k1, k2, . . . , kn), its mixed Cayley-Sylvester resultant matrix (m = n) looks like, G =         G0 . . . ... Gnk1 G0 ... . . . Gnk1        

(n+1)k1×k1

where Gi are submatrices with size of (n − 1)! n

i=2 ki × (n +

1)(n − 1)! n

i=2 ki.

Transformation matrices

Sylvester type S (n + 1)! n

i=1 ki × (n + 1)! n i=1 ki

↓ T Mixed C-S type G

(n+1)! m

n

i=1 ki × (n+1)! m

n

i=1 ki

↓ E Cayley type C n! n

i=1 ki × n! n i=1 ki

Some open problems

since we only deal with the generic n-degree polynomial system, there are many questions about specific polynomial system the singularity of resultant matrix the extraneous factor problem computational complexity etc.

• !

Weikun Sun

School of Science Tianjin University of Technology and Education Tianjin, 300222, P. R. China

sunweikun@tute.edu.cn