The Geometry of Polynomial Division and Elimination

Introduction
Multivariate Polynomial Division
Elimination
Conclusions
The Geometry of Polynomial Division and
Elimination
Kim Batselier, Philippe Dreesen
Bart De Moor
Katholieke Universiteit Leuven
Department of Electrical Engineering
ESAT/SCD/SISTA/SMC
May 2012
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Outline
1
Introduction
2
Multivariate Polynomial Division
3
Elimination
4
Conclusions
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Symbolic Methods
Computational Algebraic Geometry
Emphasis on symbolic methods
Computer algebra
Huge body of literature in Algebraic Geometry
Wolfgang Gröbner
(1899-1980)
Bruno Buchberger
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Changing the Point of View
Richard Feynman
Seeing things from a Linear Algebra perspective
Is it possible to use Linear Algebra instead?
New insights/interpretations?
New methods?
Numerical Algebraic Geometry
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Research on Three Levels
Conceptual/Geometric Level
Polynomial system solving is an eigenvalue problem!
Row and Column Spaces: Ideal/Variety ↔ Row space/Kernel of M ,
ranks and dimensions, nullspaces and orthogonality
Geometrical: intersection of subspaces, angles between subspaces,
Grassmann’s theorem,. . .
Numerical Linear Algebra Level
Eigenvalue decompositions, SVDs,. . .
Solving systems of equations (consistency, nb sols)
QR decomposition and Gram-Schmidt algorithm
Numerical Algorithms Level
Modified Gram-Schmidt (numerical stability), GS ‘from back to front’
Exploiting sparsity and Toeplitz structure (computational complexity
O(n2 ) vs O(n3 )), FFT-like computations and convolutions,. . .
Power method to find smallest eigenvalue (= minimizer of polynomial
optimization problem)
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Polynomials as Vectors
Graded Xel Ordering
Let a and b ∈ Nn0 . We say a >grxel b if
|a| =
n
X
i=1
ai > |b| =
n
X
bi , or |a| = |b| and a >xel b
i=1
where a >xel b if, in the vector difference a − b ∈ Zn , the leftmost
nonzero entry is negative.
Examples
(2, 0, 0) >grxel (0, 0, 1) because |(2, 0, 0)| > |(0, 0, 1)| which
implies x21 >grxel x3
(0, 1, 1) >grxel (2, 0, 0) because (0, 1, 1) >xel (2, 0, 0) which
implies x2 x3 >grxel x21
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Polynomials as Vectors
Vector Representation
Defining a monomial ordering allows a vector representation
Each column of the vector corresponds with a monomial,
graded xel ordered and ascending from left to right
LM(p) , Leading Monomial of polynomial p according to
monomial ordering
Example: the polynomial 2 + 3x1 − 4x2 + x1 x2 − 7x22 is
represented by
1
2
x1
3
x2
−4
x21
0
x1 x2
1
x22
−7
Cdn : vector space of all polynomials in n indeterminates with
complex coefficients up to a degree d
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Outline
1
Introduction
2
Multivariate Polynomial Division
3
Elimination
4
Conclusions
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Definition Divison
Definition
Fix any monomial order > on Cdn and let F = (f1 , . . . , fs ) be a
s-tuple of polynomials in Cdn . Then every p ∈ Cdn can be written as
p = h1 f1 + . . . + hs fs + r
where hi , r ∈ Cdn . For each i, hi fi = 0 or LM(p) ≥ LM(hi fi ), and
either r = 0, or r is a linear combination of monomials, none of
which is divisible by any of LM(f1 ), . . . , LM(fs ).
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Divisor Matrix
Divisor Matrix D in Cdn
Given a set of polynomials f1 , . . . , fs ∈ Cdn , each of degree di (i = 1 . . . s) and
a polynomial p ∈ Cdn of degree d then the Divisor matrix D is given by


f1
x f 
 1 1


 x2 f1 


 . 
 . 
 . 
 k 
1

D = 
xn f1 
 f 
 2 


 x1 f2 


 . 
 . 
 . 
xkns fs
where each polynomial fi is multiplied with all monomials xαi from degree 0
up to degree ki = deg(p) − deg(fi ) such that xαi LM(fi ) ≤ LM(p).
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Divisor Matrix
Example
Let p = 4 + 5x1 − 3x2 − 9x21 + 7x1 x2 and
F = {−2 + x1 + x2 , 3 − x1 }. The Divisor Matrix is then
1
f1
−2
x1 f1 
 0
D = f2 
 3
x1 f2  0
x2 f2
0

x1
1
−2
−1
3
0
x2
1
0
0
0
3
x21 x1 x2

0
0
1
1 

0
0 

−1
0 
0
−1
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Divisor Matrix
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Divisor Matrix
Divisor Matrix D
row space of D , D : all polynomials
LM(p) ≥ LM(hi fi )
P
i hi fi
s.t.
dim(D) = rank(D)
[p]D = {r ∈ Cdn : p − r ∈ D}
Set of all these equivalence classes (remainders) is denoted by
Cd /D
dim(Cd /D) = nullity(D)
Any monomial basis of a vector space R such that R ∼
= Cd /D
and R ⊂ Cdn = a normal set
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Divisor Matrix
R
r
r
p
P
i hi fi
D
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Division Algorithm
Algorithm: Multivariate Polynomial Division
Input: polynomials f1 , . . . , fs , p ∈ Cdn
Output: h1 , . . . , hs , r
D ← Divisor matrix for p
D ← linear independent rows of D
col ← indices of linear dependent columns of D
R ←Pcanonical basis of monomials corresponding with col
q = si hi fi ← project p along R onto D
r ←p−q
h = h1 , . . . , hs ← solve hD = q
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Division Algorithm
Oblique Projection
p = h1 f1 + . . . + hs fs + r with hi fi ∈ D and r ∈ R
Ps
i hi fi is found by projecting p oblique along R onto D
s
X
hi fi = p/R⊥ [D/R⊥ ]† D
i=1
p/R⊥ , D/R⊥ orthogonal complements of p orthogonal on R
and D orthogonal on R respectively
r is then found as r = p − hf
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Non-uniqueness of quotients
Non-uniqueness of quotients
General case D not of full row rank
Linear independent rows of D form a basis of D
Definition does not provide extra constraints to pick out a
certain basis
Non-uniqueness of remainders
General case D not of full column rank
Linear dependent columns of D form a monomial basis of R
Definition does provide extra constraint but still not-unique
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Implementation
Implementation
determine: rank(D), basis for D and kernel
from kernel determine the monomial basis for R
compute the oblique projection (exploiting the structure)
sparse multifrontal multithreaded rank-revealing QR
decomposition
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Outline
1
Introduction
2
Multivariate Polynomial Division
3
Elimination
4
Conclusions
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Macaulay Matrix
Macaulay Matrix
Given a set of multivariate polynomials f1 , . . . , fs ∈ Cdn , each of
degree di (i = 1 . . . s) then the Macaulay matrix of degree d is
given by


f1


 x1 f1 




..


.


xd1 −d f 
1
 n
M (d) = 

 f2 


 x1 f2 




..


.


d
−d
s
xn fs
where each polynomial fi is multiplied with all monomials up to
degree d − di for all i = 1 . . . s.
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Elimination
Elimination Problem
Given a set of multivariate polynomials f1 , .P
. . , fs ∈ Cdn and
xe ( {x1 , . . . , xn }. Find a polynomial g = si hi fi in which all
monomials xe are eliminated.
Solution
g lies in the intersection of two vector spaces:
Md = row space of M (d)
Ed = vector space spanned by monomials {x1 , . . . , xn } \ xe
containing polynomials up to degree d
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Elimination
Ed
g
o
Md
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Elimination
Elimination Algorithm
Input: polynomials f1 , . . . , fs ∈ Cdn , monomial set xe
Output: g ∈ Md ∩ Ed
d ← max(deg(f1 ), deg(f2 ), . . . , deg(fs ))
g ← []
while g = [ ] do
E(d) ← canonical basis for Ed
M (d) ← Macaulay matrix of degree d
if Md ∩ Ed 6= ø then
g ← element from intersection
else
d←d+1
end if
end while
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Elimination
Implementation
Use canonical angles between vector spaces to determine the
intersection
Qm , Qe orthogonal bases for Md and Ed
Qm QYe = Y CZ T with C = diag(cosθ1 , . . . , cosθk )
link with Cosine-Sine decomposition
Need orthogonal basis for Md : sparse rank-revealing QR
Implicitly Restarted Arnoldi Iterations to determine the
canonical angle and g
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Conclusions
Conclusions
Polynomial division: vector decomposition
Elimination: intersection of vector spaces
Oblique projections
Principal angles and CS decomposition
Sparse structured matrices
Applicable on many other problems:
approximate GCD
polynomial system solving
ideal membership problem
...
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Introduction
Multivariate Polynomial Division
Elimination
Conclusions
Conclusions
Thank You
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