A GENERALIZATION OF LUCAS ’ THEOREM TO VECTOR SPACES

The classical Lucas’ theorem on critical points of complex-valued polynomials has been generalized (cf. 1]) to vector-valued polynomials defined on K-inner product spaces. In the present paper, we obtain a generalization of Lucas’ theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] in K-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims. KEY WORDSAND PHRASES: Abstract polynomials and their pseudo-derivatives, (supergeneralized) circular regions, K-inner product spaces. 1991 AMS SUBJECT CLASSIFICATION CODES. Primary 30C1, Secondary 30C15.

Throughout, unless mentioned otherwise, E and V denote vector spaces over an algebraically closed field ofcharacteristic zero and " denotes the family of all nonconstant polynomialsP: E V. The concept of Lucas-sets for the family ", when E is a K-inner product space, was introduced in [1] and it was shown that every memberA of the familyD(E.)ofallgeneralized circular regions orE., with A, is a Lucas-set for ".This fact naturally raises two questions: Firstly, does D(Ew) exhaust all Lucas-sets in E. when E is a K-inner product space?Secondly, does there exist an analogous family of Lucas-sets for/," when E is, in general, a vector space?In this paper, we introduce the family D'(E.) of supergeneralized circular regions orE,.which answers the first question negatively and the second question affirmatively.We employ this family to generalize (to vector-valued abstract polynomials in vector spaces) the classical Lucas'   theorem on the zeros of the derivative of a polynomial and a theorem due to Marden on linear combinations of a polynomial and its derivative.
2. PRELIMINARIES.Walsh [2] has shown that the well-known Lucas' theorem (cf.[3] or [4, Theorem (6,1)]) is equivalent to the following result [4, Theorem (6.2)], namely: Any convex circular region which contains all the zeros of a complex-valued nonconstant polynomial falso contains all the zeros of the derivative f off.In terms of the terminology of Lucas-sets (cf. [1,p. 832]) this result equivalently states that convex circular regions in the complex plane are Lucas-sets for the family of all nonconstant polynomials.Our aim in this paper is to generalize the said Lucas' theorem and to investigate possible Lucas-sets for the family of all vector-valued abstract polynomials (cf.[1], [5]- [7]) defined on vector spaces E of arbitrary dimension.A detailed analogous study of this problem, in the special case when E is a K-inner product space, has already been made in a paper due to the author (cf.[1, pp.845-847] for a precise statement about Lucas-sets for The details in the remainder of this section can all be found in [1, pp. 833-835,839-843], apart from other alternate sources cited for completeness.E and V denote vector spaces of arbitrary dimension over an algebraically closed field K of characteristic zero.We write E,= E t3{o} and K,.K t3{oo}, where w (resp.oo) is an element having the properties of vector (resp.scalar) infinity (cf.also [8, pp. 352,372] or [9, p. 116]); Ko denotes a maximal ordered subfield of K with K0o as the set of all non-negative elements ofKo, so that (cf.[10, pp. 38-40], [11, p. 56], or [12, pp.248-255])K Ko(i) {a +ib ]a,b E Ko}, where -i 2 1.Consequently, if K0 R (the field of reals) then K C (the field of complex numbers).For z E K, the definitions of ', Re z, Im z and z are defined as in C. Similarly, the concept of K-inner product spaces (briefly written K-i.p.s.) (E, (.,.)) and the notions of Ko-convexity and Ko-normed vector spaces (E, [[) are defined likewise in C (cf. [13, pp.120-121]).If (E,(.,.)) is a K-i.p.s. then the Ko-norm on E, given by x (x,x) in for x E E, defines for each b E the mapping pb (with the tacit assumption that x /II xll equals co or 0 according as x is 0 or o) which, in turn, defines the family D(E,) of all generalized circular regions (briefly g.c.r.) of E,o [1, pp.834-835].The empty set , E, E,, and the singletons {x} (and E,o-{x}) for x E are trivial members of D(E,), whereas the family Bs(E .) of all generalized balls is rich in nontrivial members ofD(E.).
The concept of abstract homogeneous polynomials is well known (see [4]- [7], [9], [11], [14]- [17]).In what follows we briefly describe abstract polynomials and their pseudo-derivatives.A mapping P E V is called an abstractpolynomial (briefly, a.p.) of degree n if for every x,y E, P(x+py) .oA'(x"y )p' whereinA(x,y) _ V are independent of p andA,(x,y) q O.The class of all nth degree a.p.'s is denoted by ' (or ,, ff V K) and, for P E ' given by (2.2), we write F(P) {h .E h , 0 ,A,(O,h , 0}. (2.3) It is known that F(P) , .Given P E /' (via (2.2)) and h .F(P), we define for each k 1,2 n, the kth pseudo-derivative P) of P by with first few being written as P, P', etc.It is known (cf.[1, Proposition 2.3 and Remark 2.4 (I)]) that P) E -t and h F(Pt for all k, and that Pk (k "1)(X) (P(kkk (X) . Iffis an (ordinary) polynomial of degree n from K to K, then fis an a.p. of degree n from K to K, F(f) K {0}, and where fit) denotes the kthformal derivative off(see [18, p. 528], [17, p. 553], or [1, p. 842]).In particular, for h 1, we see that f)-fit) and the two notions coincide.Furthermore, if K-C, then fl becomes precisely the kth derivative fit) off as defined via calculus.For k 1, we have f -f'.

SUPERGENERALIZED CIRCULAR REGIONS
The study in [1] has revealed that the g.c.r.'s orE.and the pseudo-derivatives of a.p.'s from E to V, respectively, are natural analogues of (classical) circular regions and derivatives of (ordinary) polynomials in the complex plane, needed to formulate Lucas' theorem in a K-i.p.s.In order to achieve such a break- through for vector spaces E, in general, one needs to develop an analogous concept of circular regions in a vector space E. To this end we introduce in this section the concept of supergeneralized circular regions and establish some general properties and examples for later use.First, we recall the definition of the family D(K**) of all generalized circular regions of K,, as originally introduced by Zervos [8, p. 353].We say that a subset A of K belongs to D(K**) if and only if either A is one of the sets , K, K** or A satisfies the following two conditions: (i) 0(A) is K0-convex for all g G K-A, where 0z)-(z-g)-for every z GKd (ii) GA irA is not/(,0-convex.Fuller details about D(K**) can be found in [18, p. 527-528].
Next, we prove that oo 6/G if G is not K0-convex.  .i together with (3.11) and the fact that V() , implies at Vb(S). is would then mean at b S, which is a contradiction. is goes to establish atS D(E,).Pan ) is now established.
x. y E 0) b aitrarily hosen.n x + py S if and only L(x) + ) 0 provided p .

LUCAS' THEOREM IN VECTOR SPACES.
In this section we prove the following main theorem on the location of the null-sets of pseudo- derivatives of abstract polynomials P E K, which generalizes to vector spaces the classical Lucas' theorem in the complex plane as well as a result due to the author [1, Theorem 3.1].As an appliation of the main theorem we also generalize to vector spaces Marden's theorem [4, Corollary (18,1)] on linear combinations of a polynomial and its derivatives.If P ',, we shall write ZCP)-{xEIPCx)-O}.THEOREIVI 4.1.IfP /', audS D'(E,,) such that o S andZ(P)C_S, then z(P')c_s v h F(P).
Since h .F(P)a nd K is algebraically closed, we can write P(x +ph)-Ak(x,h)p V pK, k-o -A,(x,h).lI[p-p(x,h)] V pEK, ./..1 where A,(x,h) and p(x,h belong to K and are independent of p such that A,,(x,h) An(O,h O.In the case under consideration, if we write p p(x,h), we see that p 0 for all j and P(x) -Ao(x,h (-1)"A,,(x,h A(n,n), P(x)-A(x,h)-(-1)"-lA,,(x,h) A(n 1,n), where A(k,n) denotes the sum of all possible products of pl, p, Pn taken k at a time.Therefore P'(x)._1/p-0.
PROOF.By Remark 2.1 f (with E K), F K {0} and Z') Z') for all h K {0}.Now eorem 4.1, along with Remark 3.1 and Proposition 3.5, immediately mish e corollary.For K-C, rolla 4.2 is eentially an proeed vemion of e (classical) Lucas' eorem (see Section 2), improvement being in Se sense at we e e family D(C.) f all g.c.r.'s of C, instead of the classical c.r.'s as ed in Lucas' theorem.Using e teinolo of Lucas-m [1, Remark 3.3], eorem 4.1 ys at eve sg.c.r.A of E( A) is a Lucas-set for a.p.'s vector spaces.paaicular when E is a K-i.p.s., the family D*(E) D(E) does not exhaust all Lucas-sets. is answers the o questions posed in the introduction.Repeated applications of eorem 4.1, together with the obseations immediately preceding Remark 2.1, give e following theorem on successive pseudo-derivatives.TEOM 4.3.P , andS D'(E) such atS andZ(P)S, en Z(P))S V hF(P), lkn-1.
In order to extend the above eorem to e class , we briefly deribe e following notions and concepts, whose details can be found in [1, pp.5-847].A subset M of Vis called supportable if, for each V-M, there exists a lienar form L( 0) on Vsuch atL() 0 but L(v) 0 for all v M. ffP is given by (2.2) and if M is a supportable subset of V, we write (e).e(e,)-{x e le(x) ), F'(P).F*(P,M)-{h E IA,(O,h)M} (4.2) Since 0 M, we observe that F'(P) C. F(P) V P ",, (4.3)Now we give the following more general vector-valued formulation of Theorem 4.3.THEOREM 4.4.If P :, S D'(E,o with o S, and if M is a supportable subset of V such that F(P) s Op and E(P) _C S, then E(PC_S h F'(P), 1 sk,n-1.
PROOF.The proof is exactly the same as that of Theorem 3.9 in 1] except that the role of Theorem 3.4 in [1 is replaced by that of Theorem 4.3 of this paper.
Let us note that the hypothesis "F'(P) s " in the above theorem is not vacuous (cf. [1,Remark 3.8]).
For V K andM K {0} the above theorem deduces Theorem 4.3 (cf.[1, Remark 3.6]) and, hence, Theorem 4.4 is a more general formulation of Lucas' theorem to vector spaces.However, Theorem 4.1 is actually the basic result employed in Theorems 4.3 and 4.4.In terms of our previous terminology [1, Remark 3.3], we see that sg.c.r.'s of E,o are indeed the Lucas-sets for vector-valued a.p.'s from E to V.
Finally, we also note that Theorems 4.1, 3.4 and 3.9 in [1] are, respectively, special cases of the present Theorems 4.1, 4.3 and 4.4 when E is a K-i.p.s.(cf.Proposition 3.5 (a)).Since the above-referred theorems in [1] cannot be generalized to vector spaces over nonalgebraically closed fields of characteristic zero (cf. 1', Example 4.1]), the same is true for our present theorems.
In the remainder of this section we discuss some interesting examples to suport the validity of hypotheses of our theorems here.In case E is a K-i.p.s. and S D(E,o)C_ D'(Eo,), this claim is supported by a number of examples discussed in [1, Section 4].We therefore discuss examples for sg.c.r.'s in vector spaces only.
EXAMPLE 4.5.Let E be an arbitrary vector space of finite or infinite dimension, with dimE 2, and consider any hyperplane S a + M0, where a E E andM0 is a maximal subspace orE.ThenS E D'(E,) by Remark 3.7 (H) and oCS.Given any fixed element v Mo (possible), every element x EE has the unique representation (cf.[1, p. 80]) x-u+tv for some uM 0, tEK.
Finally, we apply Theorem (4.1) in a different direction to generalize to vector spaces a result due to Marden [4, Corollary (18,1)] on linear combinations of a polynomial and its derivative.In what follows, S + a denotes the set (s + a Is S } where S _C E and a E. TEOREM 4.7.GivenP C'n and ..K, defineR(x) P(x)-LP'(x) forh F(P).IfS .D'(E,o), taS, such that Z(P C_ S, then Z(R C S O(S + n' ).
PROOF.The proof is obvious for k 0. Therefore, we assume that 0 and R (x) 0. If x C S, we are done.Ifx S then by Theorem 4.1P',(x)O and P(x), 0 (since Z(P C_ S).Therefore (cf. the first equation in (4.1)), 11% 1 0 .