MATRIX COMPLETION PROBLEMS IN MULTIDIMENSIONAL SYSTEMS

1. MATRIX COMPLETION PROBLEMS IN MULTIDIMENSIONAL SYSTEMS wlawton@math.nus.edu.sg Wayne M. Lawton Dept. of Mathematics, National University of Singapore Block S14, Lower Kent Ridge Road, Singapore 119260 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 commutative ring with identity is unimodular if there exists such that is unimodular if is a Hermite ring if every unimodular row vector is the first row of a unimodular matrix (completion)

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

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

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

8. CONTINUOUS FUNCTIONS Proof Let be the second row of a matrix completion of Since and linearly independent at each point, hence Multiply the second and third rows of by to obtain Hopf’s theorem implies there exists a homotopy Choose Construct where satisfies and if M is continuous and completes P.

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

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

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

12. WEIRSTRASS p-FUNCTION Define where is stereographic projection and is periodic and defines

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

14. RESULTS Theorem 2. is isomorphic to the ring And therefore is not a Hermite ring. Furthermore the ring is not a Hermite ring. Proof Define the map by Results for p imply that is 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 THAT DOES NOT ADMIT A MATRIX COMPLETION Proof Compute so these maps are never antipodal, hence

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