Systems & Control Letters 2 (1982) 222-224 December 1982 North-Holland Publishing Company

Polynomial rings over more variables are not

arbitrary fields in two or pole assignable

Allen T A N N E N B A U M

Center for Mathematical System Theory, Department of Mathe- matics, University of Florida, Gainesville, FL 32611, USA and Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel

Received 20 July 1982 Revised 25 September 1982

An explicit example of a completely reachable pair of 2 × 2 matrices with entries in the ring k[x,y], k an arbitrary field, is written down which is not pole assignable, i.e. the ring k[x,y] is not pole assignable.

Keywords: Pole assignability, Elliptic curve, Complete reacha- bility, Projective ideal, Branch point.

In Section l, we construct an explicit example which shows that k[x, y] (we denote the variables in case N = 2 by x and y ) is not pole assignable for k arbitrary. In Section 2, we give the derivation based on the work of [5].

We should mention that in their fundamental paper [1], Bumby et al. show that the ring R[x, y] is not pole assignable using a clever topological argument. Our construct ion of course works over the real numbers and gives a completely algebraic counterexample. Finally, by a result of Morse [3] our example cannot be improved to N = 1, i.e. k[x] is pole assignable.

1. The example

Introduction

In our previous paper [5] using some elemen- tary properties of elliptic curves we constructed an example of a completely reachable pair of matrices (F, G) defined over

k [ X , . . . . . XN], U>_-2,

k an algebraically closed field, which is not pole assignable. Now the pair of matrices we got were derived as linear operators coming from an ab- stract module, and several people have asked us if we could write down the pair explicitly. This is the purpose of the present note.

Actually we can improve somewhat on our pre- vious example here. First of all, the counterexam- ple of [5] consisted of a pair of matrices (F, G) with F 2 × 2, and G 2 × 4. We can now reduce G to a 2 x 2 matrix. Secondly we will show that the example shows that

k [ X , . . . . . XN], U>~2

is not pole assignable for k an arbitrary field (not necessarily algebraically closed). To keep things simple we will take the case N = 2 (the general case is identical).

In this section we write down an explicit exam- ple showing that k[x, y] is not pole assignable for k arbitrary. In order to do this we will need the following lemma which is a simple corollary of the proof of Proposit ion (3.3) of [1] and a generaliza- tion of (2.1) of [5] to the case of an arbitrary field k:

Lemma 1.1. Let (F, G) be pole assignable where the matrices F and G have entries in the polynomial ring k[x I . . . . . XN], k an arbitrary field. Then Im G (the image of the linear map G) contains a vector v(x I . . . . . XN) such that

v ( x ° . . . . . x ° ) = 0

for all (x ° . . . . . x ° ) ~ k u were k is the algebraic closure of k.

Proof. We derive the result directly from Proposi- tion 3.3 of [1] using an argument suggested to us by E. Sontag via a private communicat ion. Indeed, the proof of 3.3 in [1] and our hypothesis imply that for every L m × n with entries in k[x ,y] , there is some unimodular element in

( F + GL) -I Im G.

222 0167-6911 /82 /0000-0000 /$02 .75 © 1982 Nor th -Hol land

Volume 2, Number 4 SYSTEMS & CONTROL LETTERS December 1982

Since (F, G) is pole assignable we can choose L such that F + GL is invertible, so that Im G con- tains a unimodular element as required. []

Example 1.2. Using the explicit elliptic curve

q(x ,y )=y2 + x3--x

and the construct ion of [5] (see Section 2 below for details) we claim that following pair of matrices (F , G) defined over the ring k[x, y] is completely reachable but not pole assignable:

F = - Y x2 1 G = xy x y y 1 - x 2

First we check the complete reachability. As is well known (see e.g. [4], p. 29) it is enough to check complete reachability pointwise for each (Xo, Yo) ~ ~.2. We can compute that

F( xo, Yo)G( xo, Yo)

= Xo'-Y°+Yo 2 - X o Y o - ( X ~ - I )

so that the reachability matrix

[ G(xo ,Y0) r(xo,Yo)CIxo,Yo)]

xo xoyo -yo - x 0 y g - ( x o

y0 l - x o xo +yg ,'o / Suppose first that yo z + Xo 3 - x 0 * 0. Then

det G(xo,Yo) = -xo(Yo 2 + x30 - Xo)~O

provided that x 0 * 0, so in this case Im G(x o, Yo) = /c 2. If x 0 = 0, then the second and fourth col- umns of the reachability matrix are linearly inde- pendent for any choice of Yo- If yo z + x 3 - x 0 = 0, then the determinant of the matrix formed by the first and third columns of the reachability matrix is Xo 3 +yo 2 = - x 0 which is non-zero provided that x 0 :~ 0. Hence in each case the rank of the reacha- bility matrix is maximal for all ( x o , Y o ) ~ ~:2, and so (F , G) is completely reachable.

Finally we check that (F, G) is not pole assig- nable using Lemma 1.1. We need to show that Im G has nowhere zero sections, i.e. every_ vector v ~ Im G has a zero (a point (xo, Yo)~ k 2 such that v (x o, Yo) = 0). This can either be done through a messy long computa t ion (as shown to the author by P. Khargonekar) , but we will use the following

argument from [5] which is much simpler (see Section 2 for the motivation behind this). Indeed set

A =k[x, y l / ( yZ + x 3 - x)

and

M = (x , y ) k [ x , y ] / ( y 2 + x ' - x ) .

Then via the explicit isomorphism

w : A O A ~ M O M

given by

w(a, b)= (ax + by, -ay + bx 2)

(see Section 2 and [5]) it is easy to show that

Im G ~ [ ...... .i A -= M.

Since M clearly has nowhere zero sections, neither does Im G (see [2]) from which we see by 1.1 that (F , G) is not pole assignable.

Hence we see that k[x,y] (and therefore k[x t . . . . . X N ] , N > 2) is not pole assignable for k an arbitrary field.

2. Derivation of the example

We would like to very briefly sketch here how the example of Section 1 was derived using the ideas of [5].

The main idea of [5] is to start with a fixed elliptic curve x which we now take to be rationally defined over the arbitrary field k and embed it in the projective plane P~2 as a cubic curve. We set X ' = X - Po where Po is an inflection k-point, and we let A = (ring of regular functions on X'). Then A -- k[x, y]/q(x, y) where q(x, y) = 0 is the equa- tion of X ' in the affine plane A~. We let p * Po be a point of X which is a branch k-point and set M = (regular functions on X' which vanish at p). Clearly M c A is a projective ideal. One then can show that M • M --- A 2 as A-modules but M m A (see [5] for details).

In [5] we constructed an example (F, G) of a completely reachable pair of matrices over k[x, y] which is not assignable as follows. We define homomorphisms

£ ' , P : M ~ M ~ M ~ M

by

F ( f , g ) : = ( O , f ) , b ( f , g ) : = ( f , O ) .

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Volume 2, Number 4 SYSTEMS & CONTROL LETTERS December 1982

Since M • M --- A • A these can be lifted to linear operators

F, P: ( k [ x , y])2 ~ ( k [ x , y ] ) 2 .

Taking

G = P

we can easily check (see [5]) that (F, G) is com- pletely reachable but not pole assignable. Actually the computation of Section 1 shows that already the pair ( F, P) (our P here corresponds to the G of Section 1) is completely reachable but not pole assignable.

The example of Section 1 was obtained by taking the explicit elliptic curve defined by

q(x , y ) = y 2 + x 3 - x=O.

One then has that

A = k[x , y ] / ( y 2 + x 3 - x ) ,

M = ( x , y ) k [ x , y ] / ( y 2 + x 3 - x ) .

Using the ideas of [5] one can compute an explicit isomorphism w : A • A ---, M • M defined by

w( a, b) = ( ax + by, - ay + bx 2)

from which one can write down the required matrices via the procedure outlined above, and these turn out to be precisely the matrices of Section 1.

References

[1] R. Bumby, E.D. Sontag, H.J. Sussmann and W. Vasconce- los. Remarks on the pole-shifting problem over rings, J. Pure and Appl. Algebra 20 ( 1981 ) 113-127.

[2] R. Hartshorne. Algebraic Geornet~., GTM 52 (Springer, Heidelberg-New York, 1977).

[3] A.S. Morse, Ring models for delay-differential systems, A utomatica 12 (1976) 529-531.

[4] E. Sontag, On split realizations of response maps over rings. lnform, and Control 37 (1979) 23-33.

[5] A. Tannenbaum, On pole assignability over polynomial rings, Systems and Control Letters 2 (1982) 13-16.

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