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 = alxl + al−1xl−1 + · · · + a1x + a0, and g = bmxm + bm−1xm−1 + · · · + b1x + b0. Res(f, g) =

• al

al−1 al−2 · · · a0 al al−1 al−2 · · · a0 ... ... ... ... al al−1 al−2 · · · a0 bm bm−1 bm−2 · · · b0 bm bm−1 bm−2 · · · b0 ... ... ... ... bm bm−1 bm−2 · · · b0

• 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: Fi(x1, . . . , xn) (i = 0, . . . , n). Res : a polynomial in the coefficients of Fi. For a given system fi (i = 0, . . . , n), Res(f0, . . . , fn) = 0 ⇐ ⇒ fi = 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, yn = δny. K: ordinary differential field of char. zero K{y} := K[δny](n ∈ N): diff. poly. ring Bj

i : All monomials of total degree less than or equal to j with the basis

• f the first i elements of B = {1, y, y1, y2}.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 8 / 26

Notations

Notations

δ: a derivation operator, yn = δny. K: ordinary differential field of char. zero K{y} := K[δny](n ∈ N): diff. poly. ring Bj

i : All monomials of total degree less than or equal to j with the basis

• f the first i elements of B = {1, y, y1, y2}.

For example, B2

2 = {1, y, y2} and B2 3 = {1, y, y1, y2, yy1, y2 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∈ms,r uMM

• ms,r: differential monomials in Y of order ≤ s and degree ≤ r
• uM: 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∈ms,r uMM

• ms,r: differential monomials in Y of order ≤ s and degree ≤ r
• uM: differential indeterminates

Differential resultant by Carr` a Ferro : For g1, g2 with ord(g1) = m, ord(g2) = n, δRes(g1, g2) is the algebraic Macaulay’s resultant of the system P(g1, g2) = {δng1, δn−1g1, . . . , g1, δmg2, δm−1g2, . . . , g2} in the polynomial ring Sm+n = K[y, δy, . . . , δm+ny] 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 g1 = a0y2

1 + a1y1y + a2y2 + a3y1 + a4y + a5,

g2 = b0y2

1 + b1y1y + b2y2 + b3y1 + b4y + b5,

where ai, bi 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

2

y 4

2 y1

. . . y 3

2

. . . y 1                     d1a0 . . . δa5 . . . . . . . . . . . . . . . δa4 δa5 . . . a0 . . . . . . . . . . . . . . . a4 a5 d2b0 . . . δb5 . . . . . . . . . . . . . . . δb4 δb5 . . . b0 . . . . . . . . . . . . . . . b4 b5                    

• B3

3δg1

• B3

3g1

• B3

3δg2

• B3

3g2 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

2

y 4

2 y1

. . . y 3

2

. . . y 1                     d1a0 . . . δa5 . . . . . . . . . . . . . . . δa4 δa5 . . . a0 . . . . . . . . . . . . . . . a4 a5 d2b0 . . . δb5 . . . . . . . . . . . . . . . δb4 δb5 . . . b0 . . . . . . . . . . . . . . . b4 b5                    

• B3

3δg1

• B3

3g1

• B3

3δg2

• B3

3g2

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 f1 = ay

d1 1 yd1

1 + ay

d1−1 1

yyd1−1 1

y + · · · + a0, f2 = by

d2 1 yd2

1 + by

d2−1 1

yyd2−1 1

y + · · · + b0, where 1 ≤ d1 ≤ d2, ay

d1 1 , . . . , a0, by d2 1 , . . . , b0 are diff. indeterminates. 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 f1 = ay

d1 1 yd1

1 + ay

d1−1 1

yyd1−1 1

y + · · · + a0, f2 = by

d2 1 yd2

1 + by

d2−1 1

yyd2−1 1

y + · · · + b0, where 1 ≤ d1 ≤ d2, ay

d1 1 , . . . , a0, by d2 1 , . . . , b0 are diff. indeterminates.

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 = BD

3 ∪ y2BD−1 3

with D = 2d1 + 2d2 − 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 = BD

3 ∪ y2BD−1 3

with D = 2d1 + 2d2 − 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(p1) = y2 yd1−1

1

, mm(p2) = yd2

1 , mm(p3) = yd1, mm(p4) = 1,

where p1 = δf1, p2 = δf2, p3 = f1, p4 = f2.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 15 / 26

Matrix formula with algebraic Macaulay resultant

• 2. Choosing main monomials

Main monomial of p: mm(p). mm(p1) = y2 yd1−1

1

, mm(p2) = yd2

1 , mm(p3) = yd1, mm(p4) = 1,

where p1 = δf1, p2 = δf2, p3 = f1, p4 = f2. Divide E into four mutually disjoint sets E = T1 ∪ T2 ∪ T3 ∪ T4, where T1 = {Y α ∈ E : mm(p1) divides Y α}, T2 = {Y α ∈ E : mm(p1) does not divide Y α but mm(p2) does}, T3 = {Y α ∈ E : mm(p1), mm(p2) do not divide Y α but mm(p3) does}, T4 = {Y α ∈ E : mm(p1), mm(p2), mm(p3) do not divide Y α}.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 15 / 26

Matrix formula with algebraic Macaulay resultant

• 3. The coefficient matrix

Write down a system of equations Y α/mm(p1) ∗ p1 = 0, for Y α ∈ T1, Y α/mm(p2) ∗ p2 = 0, for Y α ∈ T2, Y α/mm(p3) ∗ p3 = 0, for Y α ∈ T3, Y α/mm(p4) ∗ p4 = 0, for Y α ∈ T4.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 16 / 26

Matrix formula with algebraic Macaulay resultant

• 3. The coefficient matrix

Write down a system of equations Y α/mm(p1) ∗ p1 = 0, for Y α ∈ T1, Y α/mm(p2) ∗ p2 = 0, for Y α ∈ T2, Y α/mm(p3) ∗ p3 = 0, for Y α ∈ T3, Y α/mm(p4) ∗ p4 = 0, for Y α ∈ T4. Suppose that the coefficient matrix of the system is denoted by Md1,d2, then the determinant of it is det(Md1,d2).

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 16 / 26

Main properties of matrix Md1,d2 Proposition 1. The coefficient matrix Md1,d2 is square.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 17 / 26

Main properties of matrix Md1,d2 Proposition 1. The coefficient matrix Md1,d2 is square. Proposition 2. det(Md1,d2) is not identically equal to zero.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 17 / 26

Main properties of matrix Md1,d2 Proposition 1. The coefficient matrix Md1,d2 is square. Proposition 2. det(Md1,d2) is not identically equal to zero. Proposition 3. det(Md1,d2) is a nonzero multiple of δRes(f1, f2).

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 17 / 26

Example revisited

Here, E = y2B4

2 ∪ B5 2 and mm(δg1) = y2 y1, mm(δg2) = y2 1,

mm(g1) = y2, mm(g2) = 1, one has a 36 × 36 square matrix

y2y 4

1

y2y 3

1 y . . . y2y 2 1 y 2 . . .

y 1

T =

                    2a0 a1 . . . . . . . . . . . . . . . . . . δa4 δa5 . . . a0 . . . . . . . . . . . . . . . a4 a5 2b0 b1 . . . . . . . . . . . . . . . . . . δb4 δb5 . . . b0 . . . . . . . . . . . . . . . b4 b5                    

• B3

3δg1

• (B3

2 ∪ y1B2 2 ∪ y2B2 2)g1

• B3

3δg2

• (B1

2 ∪ y1B1 2 ∪ y2B1 2)g2 Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 18 / 26

Matrix formula with algebraic sparse resultant

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 19 / 26

Matrix formula with algebraic sparse resultant

The structure of the considered system f1 = ay

d1 1 yd1

1 + ay

d1−1 1

yyd1−1 1

y + · · · + a0, ց ց δf1 = (d1ay

d1 1 yd1−1

1

y2 + δay

d1 1 yd1

1 ) + · · · + δa0,

Similar for f2.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 20 / 26

Matrix formula with algebraic sparse resultant

Construction of the matrix

• 1. Column monomials set E: Lifted method

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 21 / 26

Matrix formula with algebraic sparse resultant

Construction of the matrix

• 1. Column monomials set E: Lifted method

Choose a perturbed vector σ = (σ1, σ2, σ3) with 0 < σi < 1 with i = 1, 2, 3, then E = BD

3 ∪ y2BD−1 3

.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 21 / 26

Matrix formula with algebraic sparse resultant

Construction of the matrix

• 1. Column monomials set E: Lifted method

Choose a perturbed vector σ = (σ1, σ2, σ3) with 0 < σi < 1 with i = 1, 2, 3, then E = BD

3 ∪ y2BD−1 3

.

• 2. Row polynomials for the monomials: Linear programming problem

to divide E = S1 ∪ S2 · · · ∪ Sn+1.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 21 / 26

Matrix formula with algebraic sparse resultant

Construction of the matrix

• 1. Column monomials set E: Lifted method

Choose a perturbed vector σ = (σ1, σ2, σ3) with 0 < σi < 1 with i = 1, 2, 3, then E = BD

3 ∪ y2BD−1 3

.

• 2. Row polynomials for the monomials: Linear programming problem

to divide E = S1 ∪ S2 · · · ∪ Sn+1. Let the lifting functions of pi be li = (Li1, Li2, Li3), i = 1, . . . , 4, where Lij are undetermined parameters.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 21 / 26

Matrix formula with algebraic sparse resultant

• Lemma. Under the conditions σ = (σ1, σ2, σ3) with 0 < σi < 1

and L11 − L12 − L21 + L22 ≤ 0, L13 ≤ L23, L21 ≤ L31 ≤ L11 ≤ L41, L22 ≤ L12 ≤ L32 ≤ L42, L31 = L32 + L41 − L42, we obtain the same main monomials as the first method.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 22 / 26

Matrix formula with algebraic sparse resultant

• Lemma. Under the conditions σ = (σ1, σ2, σ3) with 0 < σi < 1

and L11 − L12 − L21 + L22 ≤ 0, L13 ≤ L23, L21 ≤ L31 ≤ L11 ≤ L41, L22 ≤ L12 ≤ L32 ≤ L42, L31 = L32 + L41 − L42, we obtain the same main monomials as the first method.

• Theorem. The sparse resultant of f1, f2, δf1, δf2 as polynomials in

y, y1, y2 is not identically zero and contains δRes(f1, f2) as a factor.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 22 / 26

Example revisited

Example revisited Let σ = (0.01, 0.01, 0.01) and l1 = (7, −4, −5), l2 = (5, −9, 5), l3 = (6, 2, 1), l4 = (8, 4, 7).

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 23 / 26

Example revisited

Example revisited Let σ = (0.01, 0.01, 0.01) and l1 = (7, −4, −5), l2 = (5, −9, 5), l3 = (6, 2, 1), l4 = (8, 4, 7). Sparse resultant of δg1, δg2, g1 and g2 in variables y, y1, y2 is |T| |S|, where S is a sub-matrix of T.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 23 / 26

Summary

Summary: Matrix formula of differential resultant with two methods:

• 1. Macaulay algebraic resultant
• 2. Sparse algebraic resultant

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 24 / 26

Summary

Summary: Matrix formula of differential resultant with two methods:

• 1. Macaulay algebraic resultant
• 2. Sparse algebraic resultant

Problems for future research: Matrix formula for general case

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 24 / 26

Conjecture

Conjecture: Let P = {f1, f2, . . . , fn+1} be n + 1 generic differential polynomials in n indeterminates, ord(fi) = si, and s = n

i=0 si.

Then the sparse resultant of the algebraic polynomial system f1, δf1, . . . δs−s0f1, . . . , fn+1, δfn+1, . . . δs−snfn+1 is not zero and contains the differential resultant of P as a factor.

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 25 / 26

Thanks➐

Zhi-Yong Zhang (AMSS, CAS) Matrix Formula of Differential Resultant 28,Oct, 2012 26 / 26