Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work Solving Systems of Polynomial Equations: Algebraic Geometry, Linear Algebra, and Tensors? Philippe Dreesen Bart De Moor Katholieke Universiteit Leuven – ESAT/SCD Workshop on Tensor Decompositions and Applications (TDA2010) Monopoli, Italy, September 2010 1 / 27
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Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Solving Systems of Polynomial Equations:
Algebraic Geometry, Linear Algebra, and Tensors?
Philippe Dreesen Bart De Moor
Katholieke Universiteit Leuven – ESAT/SCD
Workshop on Tensor Decompositions and Applications (TDA2010)Monopoli, Italy, September 2010
1 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Outline
1 Univariate Polynomials
2 Multivariate Polynomials
3 Applications
4 Tensors
5 Conclusions and Future Work
2 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Outline
1 Univariate Polynomials
2 Multivariate Polynomials
3 Applications
4 Tensors
5 Conclusions and Future Work
3 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Polynomials, Matrices and Eigenvalue Problems
Characteristic Polynomial
The eigenvalues of A are the roots of
p(λ) = det(A− λI) = 0
Companion Matrix
Solvingq(x) = 7x3
− 2x2− 5x + 1 = 0
leads to
0 1 00 0 1
−1/7 5/7 2/7
1xx2
= x
1xx2
4 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Sylvester Matrix
Sylvester Resultant
Consider two polynomials f(x) and g(x):
f(x) = x2− 3x + 2
g(x) = x3− 4x2
− 11x + 30
Common roots iff S(f, g) = 0
S(f, g) = det
2 −3 1 0 00 2 −3 1 00 0 2 −3 1
30 −11 −4 1 00 30 −11 −4 1
James Joseph Sylvester
5 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
evaluate the vector containing the powers of x at x⋆ = 2
6 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Sylvester Matrix
Find a vector in the nullspace of the Sylvester matrix,
2 −3 12 −3 1
2 −3 130 −11 −4 1
30 −11 −4 1
−0.0542−0.1083−0.2166−0.4332−0.8664
= 0
normalize such that the first entry equals 1:
2 −3 12 −3 1
2 −3 130 −11 −4 1
30 −11 −4 1
124816
= 0
7 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Conclusion: Main Ingredients
Linear Algebra turns out to be suitable framework
Main Ingredients:
Linearize problem by separating coefficients and monomialsSolutions live in the nullspace of coefficient matrixExploit structure in monomial basisEigenvalue problems
Multivariate case?
8 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Outline
1 Univariate Polynomials
2 Multivariate Polynomials
3 Applications
4 Tensors
5 Conclusions and Future Work
9 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work
Outline of Algorithm
Macaulay: multivariate Sylvester construction
Linearize by separating coefficients and monomials
Algorithm:
1 Build coefficient matrix M
2 Find basis for nullspace of M
3 Find solutions from eigenvalue problem
Etienne Bezout James Joseph Sylvester Francis Sowerby Macaulay
10 / 27
Univariate Polynomials Multivariate Polynomials Applications Tensors Conclusions and Future Work