CONTOUR INTEGRALS AND THEIR APPLICATIONS Wayne Lawton Department of Mathematics National University of Singapore S ,

ARYABHATA characterized the set { (x, y) } of integer solutions of the equation where a and b are integers. Clearly this equation admits a solution if and only if a and b have no common factors other than 1, -1 (are relatively prime) and then Euclid’s algorithm gives a solution. Furthermore, if (x,y) is a solution then the set of solutions is the infinite set Van der Warden, Geometry and Algebra in Ancient Civilizations, Springer-Verlag, New York, 1984.

BEZOUT investigated the polynomial version of this equation Bezout identities in general rings arise in numerous areas of mathematics and its application to science and engineering: Clearly this equation has a solution iff common roots and then Euclid’s algorithm gives a solution. have noand Algebraic Polynomials: control, Quillen-Suslin Theorem Laurent Polynomials: wavelet, splines, Swan’s Theorem H_infinity: the Corona Theorem Entire Functions: distributional solutions of systems of PDE’s Matrix Rings: control, signal processing E. Bezout, Theorie Generale des Equations Algebriques, Paris, 1769.

INEQUALITY CONSTRAINTS areTheorem If RPLP LP Proof. Let LP on the unit circle thenandon real onwith Choose a LP that is real onwith then choose W.Lawton & C.Micchelli, Bezout identities with inequality constraints, Vietnam J. Math. 28:2(2000) W.Lawton & C.Micchelli, Construction of conjugate quadrature filters with specified zeros, Numerical Algorithms, 14:4 (1997)

UPPER LENGTH BOUNDS Theorem There existswith Proof: Uses resultants. Furthermore, for fixed and for fixed L

LOWER LENGTH BOUNDS Theorem Proof: See VJM paper. andwith For any positive integer n, there exist LP Question: Are there better ways to obtain bounds that ‘bridge the gap’ between the upper and lower bounds

CONTOUR INTEGRAL representation for the Bezout identity is given by Theorem Letare a disjoint contours and the contains all roots ofand then forexcludes all roots of Proof Follows from the residue calculus. where interiorof are LP, real on T, and satisfy the Bezout identity.

SOLUTION BOUNDS Lemmawhere a contour that is disjoint from and whose (annular) interior contains T

CONTOUR CONSTRUCTION on T, hence if-invariant contours then it sufficesare to consider these quantities inside of the unit disk D. For k=1,2 letunion of open disks of radius centered at zeros ofin D Since andbe the disk of this radius centered at 0. if else Theorem

CONCLUSIONS AND EXTENSIONS The contour integral method provide sharper bounds for and therefore for B than the resultant method but sharper bounds are required to ‘bridge the gap’. Contour integrals for BI with n > terms are given by whereencloses all zeros of T except for those of Residue current integrals give multivariate versions