Methods for Solving Systems of Polynomial Equations

1. Methods for Solving Systems of Polynomial Equations Zuzana Kúkelová

2. Motivation • Many problems in computer vision can be formulated using systems of polynomial equations • 5 point relative pose problem, 6 point focal length problem • Problem of correcting radial distortion from point correspondences • systems are not trivial => special algorithms have to be designed to achieve numerical robustness and computational efficiency Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

3. Polynomial equation in one unknown - Companion matrix • Finding the roots of the polynomial in one unknown, is equivalent to determining the eigenvalues of so-called companion matrix • The companion matrix of the monic polynomial in one unknown x is a matrix defined as • The characteristic polynomial of C(p)is equal to p • The eigenvalues of some matrix A are precisely the roots of its characteristic polynomial Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

4. System of linear polynomial equations- Gauss elimination • System of n linear equations in n unknowns can be written as • We can represent this system in a matrix form Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

5. System of linear polynomial equations- Gauss elimination • We can rewrite this • Perform Gauss-elimination on the matrix A => reduces the matrix A to a triangular form • Back-substitution to find the solution of the linear system Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

6. System of m polynomial equations in n unknowns (non-linear) • A system of equations which are given by a set of m polynomials in nvariables with coefficients from • Our goal is to solve this system • Many different methods • Groebner basis methods • Resultant methods Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

7. System of m polynomial equations in n unknowns – An ideal • An ideal generated by polynomials is the set of polynomials of the form: • Contains all polynomials we can generate from F • All polynomials in the ideal are zero on solutions of F • Contains an infinite number of polynomials • - generators of Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

8. Groebner basis method w.r.t. lexicographic ordering • An ideal can be generated by many different sets of generators which all share the same solutions • Groebner basis w.r.t. the lexicographic ordering which generates the ideal I = special set of generators which is easy to solve - one of the generators is a polynomial in one variable only Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

9. Groebner basis method w.r.t. lexicographic ordering • A system of initial equations - initial generators of I • This system of equations can be written in a matrix form • M is the coefficient matrix • X is the vector of all monomials • - is a monomial Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

10. Groebner basis method w.r.t. lexicographic ordering • Compute a Groebner basis w.r.t. lexicographic ordering • Buchberger’s algorithm ~ Gauss elimination • Generator - polynomial in one variable only • Finding the roots of this polynomial using the companion matrix • Back-substitution to find solutions of the whole system Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

11. An Analogy Gauss elimination Back-substitution System of equations in triangular form One polynomial equation in one unknown Solving system of linear equations Solutions Companion matrix + Back-substitution Buchberger’s algorithm Groebner basis w.r.t lexicographic ordering - One polynomial equation in one unknown Solving system of polynomial equations Solutions Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

12. System of m polynomial equations in n unknowns – An action matrix • For most problems - “Groebner basis method w.r.t. the lexicographic ordering” is not feasible (double exponential computational complexity in general) • Therefore for some problems • A Groebner basis G under another ordering, e.g. the graded reverse lexicographic ordering is constructed • The properties of the quotient ring can be used • The “action” matrix of the linear operator of the multiplication by a polynomial is constructed • The solutions to the set of equations - read off directly from the eigenvalues and eigenvectors of this matrix Polynomial division Polynomial Equations Gröbner Basis Action Matrix Solutions Buchberger’s algorithm Eigenvectors Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

13. An Analogy Compute Companion Matrix Finding the Eigenvalues of the Companion Matrix Solving one Polynomial Equation in one Unknown Solutions Requires a Groebner Basis for Input Equations Compute Action Matrix in Quotient Ring A (Polynomials modulo the Groebner basis) Finding the Eigenvalues of the Action Matrix Solving System of Polynomial Equations Solutions Zuzana Kúkelovákukelova@cmp.felk.cvut.cz