on pole assignability over polynomial rings

Volume 2, Number 1 SYSTEMS & CONTROL LETTERS

On pole assignability over polynomial rings

Allen TANNENBAUM

Department 01 Theorerical Marhemarics, The Weirmann Insrifute of Science, Rehooot 76100, Israel

all of which may be found in Hartshorne [3]. For the sake of completeness we collect some of the more important facts in Section 1.

Received 8 January 1982

1. Some remarks about elliptic curves By an explicit counterexample it is shown that the poly-

nomial ring in more than one variables over an arbitrary algebraically closed field is nor pole assignable. That is, one can

find a pair of matrices over the ring which is completely reachable but not pole assignable.

Throughout this paper k will denote a fixed algebraically_closed field of arbitrary characteristic, and all our schemes will be defined over k.

Keywords: Pole assignment, Reachability.

Introduction

Definition 1.1. An elliptic curve is a complete non-singular irreducible algebraic curve of genus 1 (see [3] for the relevant definitions).

In their paper [2], page 122, Bumby et al. raise the question if C[x,y] (the polynomial ring in two variables over C) is pole assignable. In this note by constructing an explicit counterexample we will show that

It is well known ([3], p. 319) that if X is an elliptic curve and p0 E X a (closed) point, then the complete linear system ]3p, ] defines an embedding of X into P2 (projective 2-space). Set

kb ,,...JJ, N22,

the polynomial ring in N variables over an arbitrary algebraically closed field k, is not pole assignable. Recall that this means one can find a pair of matrices (F, G) with entries in k[x,,...,x,] with F n X n, G n X m which is completely reachable, i.e. if we set

R:= k[x ,,..., x,,,],

(where A2 is affine 2-space).

then the homomorphism

1. GFG ... F”-‘G] : R”” + R”

is surjective, but which is not pole assignable, i.e. we can find elements (Y,, . ..,a, E R such that for no matrix K m X n with entries in R will we have that

Let Pit X’ denote the group of isomorphism classes of invertible sheaves over X’, and Pic’X the group of isomorphism classes of invertible sheaves over X of degree 0. Then it is well known ([3], p. 321-325, [4], p. 196) that if we choose a base point po, ( X,po) may be given a natural structure of an algebraic group, and the map X- Pit’ X given by p-0,( p. - p) defines an isomorphism of the groups (X,p,) and Pit’ X. (Actually the map is usually given as PH(~~( p -po). How- ever, since PicOX 2 PicOX via 0,(0)1+0,(-D), the map we have defined also gives an isomorphism.) Moreover we have the following:

Lemma 1.2. Pit’ X G Pit X’ uia

Ox(D-(degD)po)~(3,(0)l.,. det(zI-F-GGK)=(z-a,) ... (z-a,).

The physical relevance of such notions is discussed in [2] and [5].

Proof. From [3], (6.5), p. 133 and (6.16) p. 145, we have an exact sequence of groups

In our construction we will need to use some elementary facts from the theory of elliptic curves, O-Z L PicXq Picx’-0

0167-69 11/82/0000-0000/$02.75 0 1982 North-Holland 13

July 1982

Volume 2, Number 1 SYSTEMS & CONTROL LETTERS July 1982

where

i(d):=ti,(dp,) (dE:Z)

and

m(flx(D)):=(3,(0)].. for@,(0)EPicX.

Now we have the degree map deg : Pit X--t Z defined by deg(o,( D)) : = degree of D, and clearly deg o i = identity, so that the exact sequence splits, i.e. Pit X’ @ Z = Pit X. The isomorphism can be given explicitly by

fJx(D)~(@,dD)lxTT deg D>

where 8,(D) E Pit X. But we have also Pit X = PicOX@ Z, via

C3,(D)w(8.(D-(degD)po),degD),

from which we see that Pit’ X; Pit X’ via

@,(D- (degD)po)~flx(D)Ixe

as required. 0

Corollary 1.3. The algebraic group (X, pO) is isomorphic to Pit x’ via the map p + 6,( -p)l x, for p E X.

Proof. Immediate from Lemma 1.2 and the discus- sion above. 0

We now assume that X is ‘ordinary’, i.e. has non-zero Hasse invariant (see Tate [6], pp. 184- 185, or [3], p. 322 and p. 332). In this case, there exists a point p E X, p Zp,, such that 2p-2p,, where “- ” denotes “is linearly equivalent to”. (For characteristic k # 2, we can find 3 such points; if characteristic k = 2 then there is only one such point; see [6], p. 185, Theorem l(b), and [4] pp. 196-197.) Then the sheaf (3,( -p)I x, is self-invertible, but non-trivial. Set

A : = {ring or regular functions on X} ,

M:= r(x’,Q(-p))

= {ring of global sections of 13,( -p)l xS}.

Then M is a projective A-module, M CA (so M is

an ideal in A), and ME3 M-A, but MrA. Note of course that A is a Dedekind domain.

Now M can be generated by two elements (this can be seen directly or see e.g. Atiyah-MacDonald [ 11, p. 99) so we get an epimorphism $J: A2 --) M in the obvious way. Set N: = ker +, and note that

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since M is projective, A2 2 N @ M. Taking the determinant isomorphism we see that A2A = N @ M, so that N@M=A. But M@M=A, which implies N - M. Hence we have M 63 M = A’. . Finally let k[x,,x,] be the coordinate ring for A2 and suppose that A =k[x,. x,1/y) (i.e. 4 is an equation for x’ in A’). Set

A,:= A[x, ,..., xN] =A@k[x, ,..., xN],

MN:= M[x, ,..., x,] = M@k[x, ,..., x/J,

for N 2 3, and let A,: = A, M,: = M. Clearly MN@‘A,V M,zA,, MN@MNzAfi, M,fA,, and A, is a quotient of the polynomial ring in N variables k[x,, . . . ,x,~] for all N 2 2.

In Section 3 we will use A, and Mhi to construct our counterexample.

2. A lemma on pole assignability

In order to construct the counterexample we will first need an elementary lemma. E. Sontag pointed out to us that this lemma may also be deduced from Proposition 3.3, p. 118 of [2], but since it is so simple we prefer to give a direct proof.

Lemma 2.1. Let (F, G) be pole assignable where the matrices F and G have entries in the polynomial ring k[x,, . . . . xN], and F is n X n, G n X m. Then Im G (the image of the finear map G) contains a vector U(X,,..., xN ) such fhaf

44 ,...,x;> #O

/or all (xp,...,x”,) E kN.

Proof. Let OL,, . . . . (Y,,, p ,,..., /3, be 2n distinct elements of k. Then by hypothesis there exist two ‘state feedback’ matrices K,, K, (m X n with entries in k[x,,...,xN]) such that

det(I/--F-GK,)=,@, (r-a,),

det(zl-F-GK,)=;fl, (z-p,).

Set R:=k[x ,, . . .,xly], and let

7: R”- R”/(Im G)

Volume 2, Number 1 SYSTEMS & CONTROL LETTERS July 1982

be the natural quotient homomorphism. Moreover define Q : = ker( v o ( CY, I - F )). Note that

Q= ker(no(a,Z- F- GK,))

>ker(cr,I-F-GGK,)=:S

and S is a module of constant rank 1, hence isomorphic to R ( q = k [ x , , . . ,xN]). If v’ is its gener- ator, then v/(x: ,..., xi)#O for all (xp ,..., xi) E kN.

Next set L: = (Y, I - F - GK, and note that Q = ker( vo L). But

detL= fi (a,-p,)#O, i= I

so L is invertible, and Q = L- ‘(Im G). Hence v= Lv’ is in Im G, and v(x~,...,x~)#O for all (xp,...,xi) E kN as required. 0

3. The counterexample

We will construct now a pair of matrices (F, G) over k[x,,. . .,xN] which is completely reachable, but not pole assignable. We use the notation of Section 1, in particular the AN-module MN con- structed there for N 2 2.

Example 3.1. Define homomorphisms

F,P:M,@M,-MNe3MN

by

md+= (OJ), mg):=(f,O).

Then F and P induce operators on Ai via the isomorphism Ai = MN @ MN, and since A, is a quotient of k[x,,...,x,], we can find linear operators

F,P:(k[x ,,..., x~])~-+(~[x ,,..., x,])’

which induce i, P on MN @ MN. Let 4(x,, x2) = 0 be the equation of X’ in A’,

and via the natural inclusion

k[x,,x,] Ck[X,J,,...,X,],

we regard q E k[x ,,..., xN]. Then define G by

G:= P ; ; . [ 1 We can easily check by Lemma 2.1 that (F, G) is

not pole assignable. Indeed,

Im G @k~.~ ,....,_ ‘ins A,v s M,vI

and so Im G has no nowhere zero section, i.e. a vector v(x,,..., xN) such that u(xp,...,x~)#O for all (xy,...,xt) E kN, since M, does not have such a section. But (F, G) is completely reachable. In- deed, it is enough to check that (F, G)(xy ,..., xi) is completely reachable for each point (xp,. . . ,xt) E k N. If (xp, xi) E X’, then q( xp, x:) # 0, and so

Im G(xp,...,xs) = k2.

If (xp,x!j) E X’, then

Im G(xp,...,xi)

c MN(xp ,..., x;) @ M,(xy ,..., x;)

is the first summand. and F( xp, . . . ,x;) moves it to the second summand, so in this case again (F, G)(.+..., xz) is completely reachable. (Note that we are setting here

M,(.&...J;)

: = M,v @+,,.....r,l

w I,..., x,]/(x,-xp, . . . . x,-x;))

and so

MN(x; ,..., x;)@MN(x; ,..., x;)

=A,(xf ,..., x;)@A,(x; ,..., xi)

1 k2

since(xp,xp)EX’,wherewedefineA,,,(xf,...,xz) in the obvious way.) This completes our example.

Acknowledgement

This work was partially done while the author was a guest at the Sonderforschungsbereich of the University of Bonn. The author wishes to thank Professor F. Hirzebruch for inviting him to come and the staff for their hospitality.

References

[I] M.T. Atiyah and 1.G. MacDonald, Inrroducrion IO Corn-

ntutalioe Algebra (Addison-Wesley, Reading. MA, 1969). [2] R. Bumby, E.D. Sontag, H.J. Sussmann. and W. Vasconce-

los, Remarks on the pole-shifting problem over rings, J. Pure Appl. Algebra 20 (1981) 113-127.

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Volume 2, Number I SYSTEMS & CONTROL LETTERS July 1982

[3] R. Hartshome, Plgebruic Geomefry, GTM 52 (Springer, Heidelberg-New York, 1977).

[4] D. Mumford and K. Suominen, Introduction to the theory of moduli. in: Proc. 5th Nordic Summer School in Math., Oslo, 1970, Algebraic Geomeny, ed. by F. Oort (Wolters-

Noordhoff, Groningen, 1972) pp. 17 l-222.

[S] ED. Sontag, Linear systems over commutative rings: A survey, Richerche di Auromotica 7 (1976) I-34.

[6] J.T. Tate, The arithmetic of elliptic curves, Inuenfiones

Math. 23 (1974) 179-206.

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on pole assignability over polynomial rings

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