MATRIX COMPLETION PROBLEMS IN MULTIDIMENSIONAL SYSTEMS

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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

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### 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

OUTLINE

1. Introduction

2. Continuous functions

3. Trigonometric polynomials

4. Stable rational functions

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)

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)

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

HOMOTOPY

are homotopic

DEFINITION

if

HOPF’S THEOREM If

then

COROLLARY

Proof. Consider

J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

CONTINUOUS FUNCTIONS

Define

Then

is unimodular

For unimodular

define

Theorem 1.

For n even, a unimodular

hence

is not Hermite since the identity function on

has degree 1 and thus cannot admit a matrix completion.

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.

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

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

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.

WEIRSTRASS p-FUNCTION

Define

where

is stereographic projection

and

is

periodic and defines

WEIERSTRASS p-FUNCTION

LAURENT EXPANSION

where

is the Eisentein series of index k for the lattice

This provides an efficient computational algorithm.

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.

EXAMPLE

EXAMPLE OF A UNIMODULAR ROW IN

THAT DOES NOT ADMIT A MATRIX COMPLETION

Proof Compute

so these

maps are never antipodal, hence

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 ?