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MATRIX COMPLETION PROBLEMS IN MULTIDIMENSIONAL SYSTEMS Wayne M. Lawton Dept. of Mathematics, National University of Singapore Block S14, Lower Kent Ridge Road, Singapore 119260 [email protected] Zhiping Lin School of Electrical and Electronic Engineering Block S2, Nanyang Avenue Nanyang Technological University, Singapore 639798 [email protected] OUTLINE 1. Introduction 2. Continuous functions 3. Trigonometric polynomials 4. Stable rational functions INTRODUCTION R is a commutative ring with identity 1 PR m m QR MR R is unimodular if there exists such that PQ 1 m m is unimodular if det M 1 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) DEGREE OF MAP OF SPHERE THE DEGREE D(f) OF CONTINUOUS f : S S n is an integer that measures the direction and number of times the function winds the sphere onto itself. S {(cos , sin )} 2 S {(cos cos , cos sin , sin )} D( k ) k D(constant ) 0 D(identity ) 1 n 1 D(antipoda l) (1) D(f h) D(f)D(h) EXAMPLES 1 n HOMOTOPY f1 , f 2 : X Y are homotopic (f1 f 2 ) if F : [0,1] X Y DEFINITION F(0,) f1 , F(1,) f 2 f1 , f 2 : S S then f f D(f ) D(f ) 1 2 1 2 COROLLARY x, f (x) f (x) f f 1 2 1 2 Proof. Consider (1 t)f tf 1 2 || (1 t)f 1 tf 2 || HOPF’S THEOREM If n n J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. CONTINUOUS FUNCTIONS Define n R R(S ) is unimodular For unimodular Then n 1 PR x, P(x) 0 P define D(P) D( P || P ||) n 1 Theorem 1. For n even, a unimodular P R admits a matrix completion D(P) 0, hence n is not Hermite since the identity function on S R has degree 1 and thus cannot admit a matrix completion. Proof CONTINUOUS FUNCTIONS Let Q be the second row of a matrix completion P M of n 1 PR .Since det M 1 and Q linearly independent at each point, hence D(P) D(Q). Multiply the second and third rows of M by - 1 to obtain D(P) D(-Q) -D(Q) 0. Hopf’s theorem implies there exists a homotopy n n n g : [0,1] S S , g(0,) P, g(1,) c S , x) 0 t 0 t L 1 x, g(t , x) g(t k k-1 Construct M M L M1 where x, M k ( x ) SO(n 1) satisfies g(t , x)M (x) g(t , x) and yM (x) y if k k k k -1 yg(t , x) yg(t , x) 0.M is continuous and completes P. k k -1 Choose TRIGONOMETRIC POLYNOMIALS Let n n Prs ( R ), Trs ( R ) be the Z n periodic symmetric continuous real-valued functions, trigonometric polynomials. Isomorphic to rings of functions on the space identifying xT n and X n obtained by - x. [-1,1] under the map x cos 2x 2 X2 homeomorphic to sphere S 2 under a map : T X 2 X1 homeomorphic to interval that is 2-1 except at {(0,0),(0,.5),(.5,0),(.5,.5)} RESULT Lemma Prs ( R ) is a Hermite ring. Proof This ring is isomorphic to the ring of real-valued functions on the interval Choose a unimodular And approximate m F R([-1,1]) F/ | F | by a continuously differentiable map H : [ 1,1] S m-1 And use parallel transporting to extend to a map M : [1,1] SO(m) WEIRSTRASS p-FUNCTION Define p : C C {} by p( z ) z [( z ) ] 2 2 2 \{0} where {m ni | m, n Z} Lemma p(z) p(w) z w or z w Proof. z ( p(z), p(z)) maps the elliptic curve curve in projective C/ isomorphically onto the cubic 2 3 space defined by the equation p 4p - g 2 p - g 3 . J. P. Serre, A Course in Arithmetic, Springer, New York, 1973, page 84. WEIRSTRASS p-FUNCTION ~ 2 2 Define p : R S 2 1 1 2 p 2 2 R C C {} S where is stereographic projection 2 1 2 ( z ) w [2u , 2v, w 2] w u v 1, z u iv and (x , x ) x ix 1 1 2 1 2 ~ is 2 periodic and defines 2 2 :T S Z WEIERSTRASS p-FUNCTION LAURENT EXPANSION p(z) z k2 (2k 1)G k z 2 where Gk 2k1 2k \{0} is the Eisentein series of index k for the lattice This provides an efficient computational algorithm. RESULTS Theorem 2. 2 Prs ( R ) is isomorphic to the ring 2 R(S ) And therefore is not a Hermite ring. Furthermore the ring 2 Trs ( R ) is not a Hermite ring. : R(S ) Prs (R ) 2 by (f ) f , f R(S ). Results for p imply that is a surjective isomorphism. Proof Define the map 2 2 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 2 3 Trs ( R ) THAT DOES NOT ADMIT A MATRIX COMPLETION F1 ( X1 , X 2 ) 2.84(G 2 - G1 ) F2 ( X1 , X 2 ) 2.51(G 3- G 4 ) F3 ( X1 , X 2 ) 10 3.55 (G 1 G 2 ) - 2.56(G 3 G 4 ) G1 cos X1 , G 2 cos X2 G 3 cos ( X1 X 2 ), G 4 cos ( X1 X 2 ) Proof Compute F 0.5 | F | maps are never antipodal, hence so these D(F) D() D((identity)) 1 OPEN PROBLEMS PROBLEM 1 If F Trs ( R ) 2 3 is unimodular and has degree zero does it admit a matrix extension ? PROBLEM 2 Is the ring 2 Ts ( R ) of symmetric trigonometric polynomials a Hermite ring ? PROBLEM 3 Is the ring 2 Tr ( R ) of real-valued trigonometric polynomials a Hermite ring ?