1. MATRIX COMPLETION PROBLEMS IN MULTIDIMENSIONAL SYSTEMS Wayne M. Lawton Dept. of Mathematics, National University of Singapore Block S14, Lower Kent Ridge Road, Singapore 119260 wlawton@math.nus.edu.sg Zhiping Lin School of Electrical and Electronic Engineering Block S2, Nanyang Avenue Nanyang Technological University, Singapore 639798 ezplin@ntu.edu.sg

2. OUTLINE 1. Introduction 2. Continuous functions 3. Trigonometric polynomials 4. Stable rational functions

3. INTRODUCTION is a Hermite ring if every unimodular row vector is the first row of a unimodular matrix (completion) is unimodular if there exists is a commutative ring with identity such that is unimodular if

4. INTRODUCTION HERMITE RINGS INCLUDE 1. Polynomials over any field (Quillen-Suslin) 2. Laurent polynomials over any field (Swan) 3. Rings of formal power series over any field (Lindel and Lutkebohment) 4. Complex Banach algebras with contractible maximal ideal spaces (V. Ya Lin) 6. Principal ideal domains eg rational integers, stable rational functions of one variable (Smith)

5. DEGREE OF MAP OF SPHERE THE DEGREEOF CONTINUOUS is an integer that measures the direction and number of times the function winds the sphere onto itself. EXAMPLES

6. HOMOTOPY are homotopic if HOPF’S THEOREM If then J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. DEFINITION COROLLARY Proof. Consider

7. Define CONTINUOUS FUNCTIONS Theorem 1. is unimodular For n even, a unimodular For unimodular Then define admits a matrix completion is not Hermite since the identity function on has degree 1 and thus cannot admit a matrix completion. hence

8. Proof CONTINUOUS FUNCTIONS and Sinceof be the second row of a matrixLet completion linearly independent at each point, hence Hopf’s theorem implies there exists a homotopy Multiply the second and third rows of byto obtain Choose Constructwheresatisfies andif M is continuous and completes P.

9. Let TRIGONOMETRIC POLYNOMIALS continuous real-valued functions, trigonometric polynomials. be theperiodic symmetric Isomorphic to rings of functions on the spaceobtained by identifyingand homeomorphic to interval under the map homeomorphic to sphere under a map that is 2-1 except at

10. Lemma RESULT Proof This ring is isomorphic to the ring of real-valued functions on the interval is a Hermite ring. Choose a unimodular And approximate by a continuously differentiable map And use parallel transporting to extend to a map

11. Define WEIRSTRASS p-FUNCTION by where Lemma Proof. isomorphically onto the cubic curve in projective maps the elliptic curve J. P. Serre, A Course in Arithmetic, Springer, New York, 1973, page 84. space defined by the equation

12. Define WEIRSTRASS p-FUNCTION whereis stereographic projection and isperiodic and defines

13. LAURENT EXPANSION WEIERSTRASS p-FUNCTION where is the Eisentein series of index k for the lattice This provides an efficient computational algorithm.

14. RESULTS is isomorphic to the ring And therefore is not a Hermite ring. Furthermore the ring Theorem 2. is not a Hermite ring. ProofDefine the map by Results for p imply thatis a surjective isomorphism. The second statement follows by perturbing a row having degree not equal to zero to obtain a unimodular row.

15. EXAMPLE EXAMPLE OF A UNIMODULAR ROW IN Proof Compute THAT DOES NOT ADMIT A MATRIX COMPLETION maps are never antipodal, hence so these

16. OPEN PROBLEMS PROBLEM 1 is unimodular and has degree zero does it admit a matrix extension ? If PROBLEM 2 PROBLEM 3 Is the ring of symmetric trigonometric polynomials a Hermite ring ? Is the ring of real-valued trigonometric polynomials a Hermite ring ?