SOME THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT NEYAMAT ZAHEER

The present paper, which is a continuation of our earlier work in Annali di Mathematica [1] and Journal Math. Seminar [2] (EE6EPIA), University of Athens, Greece, deals with the problem of determining sufficiency conditions for the nonvanishing of generalized polars (with a vanishing or nonvanishing weight) of the product of abstract homogeneous polynomials in the general case when the factor polynomials have been preassigned independent locations for their respective nullsets. Our main theorems here fully answer this general problem and include in them, as special cases, all the results on the topic known to date and established by Khan, Marden and Zaheer (see Pacific J. Math. 74 (1978), 2, pp. 535-557, and the papers cited above). Besides, one of the main theorems leads to an improved version of Marden’s general theorem on critical points of rational functions of the form flf2"’’fp/fp+l’’’fq f’l being complex-valued polynomials of degree n..1


INTRODUCTION.
A few years ago, the concept of generalized polars of the product of abstract homogeneous polynomials (a.h.p.) was introduced by Marden [3] while in his attempt to generalize to vector spaces a theorem due to Bcher [4].His formulation involves the use of hermitian cones [5], a concept which was first used by HSrmander [6] in obtaining a vector space analogue of Laguerre's theorem on polar-derivatives [7] and, later, employed by Marden  [3], [8], in the theory of composite a.h.p.'s.In all these areas the role of the class of hermitian cones has been replaced by a strictly larger class of the so-called circular cones.This was successfully done byZaheer [9], [I0], [11] and [5] in presenting a more general and compact theory which incorporates into it the various independent studies made by Hormander, Marden and Zervos.A complete account of the work to date on generalized polars, which fall in the category of composite a.h.p.'s in the wider sense of the definition of the latter now in use (cf. [5],[12], [13], [14]) can be found in the papers due to Marden [3], Zaheer [5], and the authors [1], [2].Generalized polars with a vanishing weight as well as the ones with a non-vanishing weight have beea considered in the first two papers, while the third (resp.the fourth) deals exclusively with the ones having a vanishing (resp.a non-vanishing) weight.But all have a common feature that the factor polynomials involved in the generalized polar of the product have been divided into two or three groups, each of which is preassigned a circular cone containing the null-sets of all polynomials belonging to that group.Our aim here is to consider generalized polars with a vanishing or a non-vanishing weight where, in general, no two factor polynomials are necessarily required to have the same circular cone in which their null-sets must lie.
In fact, we take up the general problem of determining sufficiency conditions for the non-vanishing of generalized polar (with a vanishing or a non-vanishing weight) where the factor polynomials have been preassigned mutually independent locations for their respective null-sets.Our main theorems fully answer this general problem and include in them, as special cases, all the corresponding results on the topic known to date and established in Marden [3], Zaheer [5] and the authors [I], [2].One of the main theorems of this paper leads to a slightly improved form of Marden's general theorem on critical points of rational functions [7]. 2. PRELIMINARIES.
Throughout we let E and V denote vector spaces over a field K of character- istic zero.A mapping e E V is called (cf.[6], [15], [16], and [9]) a vector-valued a.h.p, of degree n if (for each x, y E) n P(sx + ty) r.A k(x,y) skt n-k k=o s, te K, where the coefficients Ak(X,y e V depend only on x and y.We shall call P an a. h.p. (resp.an algebra-valued a. h.p. if V is taken as K (resp.an algebra).
We denote by P the class of all vector-valued a.h.p.'s of degree n from E to V n (even if V is an algebra) and by P the class of all a.h.p's of degree n from E n to K. The nhpo of P is the unique symmetric n-linear form P.(.x ,x 2,... ,x n) from E n to V such that P(x,x,...,x) P(x) for all x : E (Hormander [6] and Hille and Phillips [15] for its existence and uniqueness).The kth polar of P, for given x l,x2,...,x k in E, is defined by P(x l,x 2 x k,x) P(x x k,x x).
Given k : K and Pk Pn k (k 1,2 ,q), we write Pl(X)P2(x)"" Pq(x)' (2.1) and Qk (x)   Pl(X)''" Pk-l(X)Pk+l(X)''" Pq(X), and define (Q-Xl,X) as an algebra-valued generalized polar of the product Q(x) q [5].The scalar r. mk is called its weight.The Term 'generalized polar' will be k=1 used in special reference to the case when V K, so as to conform with the existing terminology [5].As in Hille and Phillips [15], if n n + n 2 + + n we re-, , q call that Q e Pn' Qk e Pn_nk and Pk(Xl,X) is an algebra-valued a.h.p, of degree nk-I in x and of degree in x I, < k _< q.Therefore, (Q:x l,x) is an algebra-valued a.h.p, of degree n-1 in x and of degree in x I.
Given a nontrivial scalar homomorphism L V K [18] and [I] and a polynomial , P e P we define the mapping LP E K by (2.4) Obviously, LP g P.
In the notations of (2.1) and (2.2) the product of the poly- n nomials

LP k
Pnk is given by LQ and the corresponding partial product (LQ)k (achieved by deleting the kth factor in the expression for LQ) is given by LQk.This immediately leads to the following REMARK 2.1.The algebra-valued generalized polar (Q;xl,x) of the product Q(x) and the generalized polar (LQ;xl,x) of the corresponding product (LQ)(x), with the same m k's satisfy the relation L((Q;x l,x)) (LQ;x l,x) for every nontrivial scalar homomorphism L on V.
If K is an algebraically closed field of characteristic zero, then we know [19] and [20] that K must contain a maximal ordered subfield K such that K where -i 2 is the unity element in K.For any element z a+ ib e K (a,b E K o) we define a ib, Re(z) (z + )/2, and Izl + (a 2 + b2) I/2 in analogy with the complex plane.
We denote by K the projective field [5] and [2 I] achieved by adjoining to K an element m having the properties of infinity, and, by D(K), the class of all generalized circular region (g.c.r.) of K.The notions of K-o convex subsets of K and of D(K) are due to Zervos [21], but the definitions and a brief account of relevant details can be found in [5].For special emphasis in the field C of complex numbers, we state the following characterization of D(C): The nontrivial g. c. r. 's of are the open interior (or exterior) of circles or the open half-planes, adjoined with a connected subset (possibly empty) of their boundary.
The g.c.r.'s of C, with all or no boundary points included, are termed as (classical) circular regions (c.r.) of In vector space E over an algeraically closed field K of characteristic zero, the terms 'nucleus', 'circular mapping' and 'circular cone' are due to Zaheer [5].where TG(X,y) {sx + ty @ o s,t g K; s/t G(x,y)}.
REMARK 2.2. (I) [I].If G is a mapping {tom N into the class of all subsets of K (so that G(x,y) may not necessarily be a g.c.r.), the resulting set E (N,G) o will be termed only a oone in E.
(II) If dim E 2, then [I0]  for some A g D(K), where Xo,Yo are any two linearly independent elements of E, with N {(Xo,Yo )} and G(xo,Yo A. (III) We remark [5] that any two (and, hence, any finite number of) circular cones can always be expressed relative to an arbitrarily selected common nucleus.
Unless mentioned otherwise, K denotes an algebraically closed field of characteristic zero, E a vector space over K, and V an algebra with identity over K.The field of complex numbers is denoted by C. We denote by L ix,y] the the subspace of E generated by elemeats x and y of E, and by L 2 ix,y] the set product L ix, y] L ix,y] (i.e. the set of all ordered pairs of elements from [x,y] ).
In this section we establish the central theorem of this paper, which gives sufficiency conditions for the non-vanishing of generalized polars having a vanishing or a non-vanishing weight and which answers the general problem mentioned in the introduction.
Apart from deducing the main theorems of the authors proved earlier in [I] and [2] the present theorem applies in the complex plane to yield an improved form of a general theorem due to Marden [7].
In the following theorem we take, without loss of generality (cf.Remark 2.2 (III)), circular cones with a common nucleus.Consistently, we shall denote by Z (x,y) the null-set of an a.h.p.P (with respect P to given elemeats x,y E), defined by Zp(x,y) {sx + ty * ols,t e K; P(sx + ty) o}.
For k 1,2, q, let Pk e Pnk and E kjot Eo(N,Gk) be circular cones in E such that ZPk(X,y) c_ TGk(X,y) for all (x,y) e N and for all k.If (Q;Xl,X is the generalized polar of the )roduct Q(x) (cf. (2.1)-(2.3))withm k > o for k < p( < q) and m k < o for k > p, then (Q;x l,x) $ 0 for all linearly independent element x,x! of E such that x E-u q E(k)o and k--I q (k) x E-u Eo u Ts(xo,Yo) where (Xo,Yo) is the unique element in k=1 2 N n [x,x I], x Xo + 6Yo and q S(Xo,Yo {0 e Kml kE=I q mk/(O Ok) I mk)/(O y/6).OkeGk(Xo,Yo)}.
k=l REMARK.Let us note that 0 m must belong to S(Xo,Yo in the case when Y/6 # m and m u q Gk(Xo,Yo).Also the hypothesis 'x n q E (k)' is necessary.For k=l k=l o otherwise, S(x ,yo would be all of K and the theorem would become uninteresting.o PROOF.Let x,x! be linearly independent elements of E such that x (u q E(k))o u Ts(xo,Yo) and x o q E (k)o where (Xo,Yo) is the unique element in [x,x I] (cf.definition of nucleus [5]).Then there exists a unique set of scalars ,B,Y, (with BY O) such that x Xo + BYo and x YXo + Yo" Obviously, the choice of x implies that a/B (u q Gk(Xo,yo )) u S(x o,yo ), due to k=l the notation in (2.5).We claim that / m.This is trivial when y/ m (since 6 o and BY 0).It is obvious also when y/ m and m belongs to q Gk(Xo,yo) However, in case y/ m and m q Gk(Xo,Yo), the definition of k=l k=l S (Xo,Yo) says that m must belong to S (Xo,Yo)" So that /B m in all cases The fact that K is algebraically closed allows us to write, for each k--1,2,...,q, n k Pk(SX + tx I) N (jk s y kt).j=l J n k 0 Since Pk (x) 3=I jk # 0 for all k, we have that for each k (I < k < q) jk for all j 1,2,...,n k.If we set Ojk Yjk/jk then, using the same technique as in the beginning of the proof of Theorem 2.5 due to Zaheer [5], we conclude that Ojk e U (Gk(Xo,Yo)) for j 1,2 ..... n k, < k < q (3I) and, further, that U(Gk(Xo,Yo)) are Ko-COnvex g.c.r.'s of K, where U is the homographic transformation [21] of Km given by U(O) (0 Y)/(-BO + a).There- fore (3.I) and the K -convexity of U(Gk(Xo,yo )) give o n k k(say) E (I/nk) Ojk U(Gk(Xo,Yo)) for k =1,2 .....q.
(3.2) j=l This implies that there exist elements Pk e Gk(Xo,Yo) such that k U(Ok) (Ok Y)/(-BOk + ) m for k=l,2 q. (33) Let us write n k k mk k l (mk/nk) 0jk V k 1,2,..,q. (3.4) j=l We now claim that I + v2 +'''+ O. First we notice that the k'S cannot vanish q simultaneously.For, otherwise, Y/ would belong to all the g.c.r.'s Gk(Xo,Yo) for k=l,2,...,q, which in turn would imply that q q E (k) x Yx + 6Yo n T G (Xo,Yo) n o k=l k k=l o This contradicts the fact that x f n q E(k)o" Therefore, in order to establish the k;1 said claim, it remains only to deal with the case when at least any two of the k'S do not vanish (since the claim is obvious otherwise).Now, with thts assumption, suppose on the contrary that I + 2 + + O. Then equations (3.3) amd (3.4) , we see that 8 0 and, consequently, the last equation can be written as q r. m k [-618 + {(a618) -'(}1(-8 k + =)] 0. k=l Therefore, q q (alS) z mkI(-8 Pk + a) (618) r. mk' k=l k=l where A sd 8'( O. Or, q 68 q l mk/(a/8 pk 8"( (kE= ink) (a/8) -'( k=l q (kE=l ink)" That is, irrespective of whether 6 0 or 6 O, we get q q z mkl(al8 pk Z mk)l(alS-'(l), k=1 k=1 where 0 k e: Gk(Xo,yo) for k 1,2,...,q, (XoYo) e N o L 2 [x,x I} and x '(Xo + 6y o.This implies that a/8 e S(Xo,Yo) and, hence, that x aXo + 8Yo Ts(Xo,Yo ), contradicting the choice of x already made.Therefore k=l k=l Since Pk(X) 0 for all k and since v + + v O, the proof is complete.q q If we take l m k 0 in the above theorem, the set S(xo,Yo) remains unchanged k=l when x.t varies freely in i[x ,yo subject to the condition that x.i In order to get a simpler and more interesting version in whicht xl varies freely over all of E it is desirable to further assume that n q E k .We do precisely o k=l this to obtain the following theorem which deals exclusively with generalized polars having a vanishing weight.q (k) x q:u "+l E -k-then there exists a unique element (x yo) e N n [X,Xl] such that o o k=l q x!_ YXo + Yo and, in the present set up, S(xo,Yo Gq+l(Xo'Yo) (since k=IE m k 0).
That is, x n q E (k)o and x (u q E (k))o u Ts(Xo,Yo ).Now the proof follows k=l k=l from Theorem 3.1.
As application of Theorem 3.1 in the complex plane we prove the following corollary which, apart from generalizing the two-circle theorem and the cross-ratlo theorem of Walsh [22] (cf.also [7], Theorems 20,1 and 22,21, improves upon Marden's general theorem on critical points of rational functions ( [7] or [23] and [24]).In the following, Z(f) denotes the set of all zeros of f.COROLLARY 3.3.For each k O,l,...,p, let fk (z) be a polynomial (from to ) o" degree n k.If C k D() such that Z(fk Ck for k 0,I p and if p Ck, then every finite zero of the derivative of the rational funon k=o fo(Z)fl(z)'''fq(z) and m k n k oraccording as k <_ q or k > q.

C 2
Then Pk is an a.h.p, of degree n k from to C such that ZPk(Xo,Y o) c_C TGk(xo,Y o) for k O,l,...,p.This is so because n k Pk(X) Pk(SX + ty o) t fk(s/t) V x (s,t) 0 (3.7) and because Z(f k) __c C k T G (Xo,Yo).Now we consider the generalized polar (Q;Xl,X k of the product Q(x) of these a.h.p.'s, with m k n k or-n k according as k < q or k > q.If we take x Xo (I,0) (so that s and t o), we see as in [5], that nk-I Pk(Xo,X) (I/nk)3Pk/S (I/nk)t f(s/t)  That is, #(Q;x ,x) 0 for all elements x (s,t) for which t # 0 and for which O p+l s/t u C k. Finally, (3.9) says that f'(s/t) 0 for all s,t e C such that t 0 k=O p+l and s/t u C k.This establishes the corollary.k=o REMARK 3.4.(I) In the special case when the g.c.r.'s C k are specialized as the closed interior or the closed exterior of circles, we claim that the above corollary reduces essentially to Theorem 21,1 of Marden [7].
This is upheld by the following arguments: If the C k are taken to be the regions OkCk(Z) < 0 of Marden's Theorem 21,1, then Lemma 21,1 of Marden [7] and the succeeding arguments p+l therein show that the region u C k in our corollary is precisely the region k=o satisfying the p+2 inequalities 21,3 in Marden's theorem.
(II) In what follows we show that Corollary 3.3 holds as such when C is re- placed by K, provided the term 'derivative' is replaced by 'formal derivative'.We n know by ( [5] or [12]) that the polynomial f'(z) k ak zk-I is called k=l the formal derivative of the polynomial f(z) ak zk from K to K and that k=o n (flf2...fn)' ).' flf2...fk_l f fk+l'''fn k--1 where the f. are polynomials [12].If we now define the formal derivative of the l ,f2_f f,)/(f2 )2 quotient fl/f2 (fl being polynomials) to be given by (fl/f2) (fl 2 then the formal derivative of the quotient f (z)f (z)...f (z)/f (z)...f (z) o q q+l p is given by equation (3.9).
(3.10) k=o k=q+l In view of the definition of the formal derivative f' (z) of a polynomial f(z) from K to K and of formal partial deruatives %P/Ss of a polynomial P(s,t) from K 2 to K [5], we can easily show that Corollary 3.3 stll holds when C is replaced by K.The proof proceeds exactly on the lines of the proof of Corollary 3.3, except only that we replace C by K all along.Let us point out that the expression (3.10) is precisely the formal derivative of the function f(z) in (3.5) and it justifies the validity of steps (3.8) and (3.9) in the proof of Corollary 3. (IV) It has been shown by Marden [7] that Walsh's cross-ratio Theorem 22,2, is a special case of Marden's general Theorem 21,1, but only in terms of closed interior or closed exterior of circles (a proper subclass of D(C)).Whereas, our Corollary 3.3   validates Walsh's theorem in terms of g.c.r.'s.In fact, applying our corollary in the set up of Walsh's theorem with C.'s taken as g.c.r.'s, we conclude that every 4 finite zero of the derivative of the function fl(z)f2(z)/f3(z) lies in u Ci, where i=I C 4   IV e CInl(p pl) + n2/(p p2) n3/(p p3) 0; Pi g Ci}.
In veiw of Lemma 4.2 of the next section, we see that C 4 Ci, and that where IV m,CI(P P3'92'Pl n2/nl;-P i C R A i Consequently, every finite zero of the derivative of the said function lies in C uC2u C3U C, where C R A {}.This shows that 6In improved ve60n 0 W0/h' cao-azut/0 (V) It may be observed that an improved form of Walsh's two-circle theorems in its complete form ([5], Corollaries 2.8 and 4.3) may also be obtained from the above corollary.To this effect we apply Corollary 3.3 in the set up of Zaheer's Corollary 4.3 [5], with the D.'s replaced by g.c.r.'s C i such that m e C n C 2, and conclude that every finite zero of the derivative of the function fl(z)/f2(z) lies We point out that, in case the C.'s are taken as the regions D. of Corollary 4.3 1 of Zaheer [5], the region C 3 is precisely D(c3,r3) (cf.notation there) and we are done.
In the case when n n 2 and C i n C 2 , the conclusion just drawn still holds, but in this case the region C 3 is empty, and we are done with Corollary 2.8 in [5]. 4. THE CASE OF ALGEBRA-VALUED GENERALIZED POLARS.
Our aim in this section is to obtain a more general formulation of Theorem 3.1 that could answer the corresponding problem for algebra valued generalized polars having an arbitrary weight.In fact, it will be shown that, whereas the main theorem of this section does include in it the main theorem of the preceding section, it also incorporates into it a variety of other known results.First, we describe some concepts and establish some results that we need in this section.We refer [17], [I]  and [2] for the following material.
A subset of M of V is called fully supportable(initially termed as 'A-supportable' by Zaheer [9]) if every point outside M is contained in some ideal maximal subspace of V which does not meet M. In other words, for every e V M, there is a unique nontrivial scalar homomorphism L on V such that L() 0 but L(v) $ 0 for every v e M [18].If M is a fully supportable subset of V, then M is a supportable subset of V (regarded as a vector space), but not conversely (for definition of supportable subsets see [6]).We remark that the complement in V of every ideal maximal subspace of V is a fully supportable subset of V.
Given P P and a fully n supportable subset M of V, we shall write, for given x,y E, Ep(x,y) {sx + ty 01s,t e K; P(sx + ty) M}.Since identity map from K to K is the only nontrivial scalar homomorphism on K, the set M K-{o} is the only fully supportable subset of K (take V K in the definition) and the corresponding set Ep(x,y), as given by (4.1), becomes the null-set Zp(x,y) of P as defined in the beginning of Section 3.
In the next few lemmas, the notation (P' PI' P2' P3 stands for the cross- ratio of an element p e K with respect to given distinct elements PI'P2'P3 e K and it designates a unique element in K(for definition and other relevant details see [5]).PROOF.In order to prove the lemma it is sufficient to show that, if u G 2 u G 3 u {m} and Pl Gi (i 1,2,3), the equation l/(p pl + A/(p p2) (I + A)I(p p3 o holds true.First, we claim that none of these equations can hold unless PI' P2' P3 are distinct elements of K This is obvious in case of (4.3) due to the definition of cross-ratio.In case of (4.2), this follows from the fact that if any two of the P.'s coincide and if (4.2) holds then all the three must coincide, contra- . Therefore, we assume that PI' P2' P3 are distinct elements of K and so we divide the proof into the following two cases: Case (i).01, 02, 03 .Case (ii).One of the Ol.'s is .In this case, let us point out that 0,01,02,0 3 are distinct elements of K with only one of the O.'s being .Therefore, the equation ( 4 The definition of cross-ratio implies that each of the equations (4.2) holds if and only if (0, 0 3 0 2,01 Cases (i) and (ii) complete our proof.
{} and there exists elements O k e G 3 for p < k < q and such that e G 2 for r < k < p Pk for k < r, Pk q q r. mklCP pl) r. ink)/( p ;). k=l k=l Therefore, q r p q E mk)/( p) Z mk/(p p) + Z mk/(O O) + Z mk/(O p) k=l k=l k=r+l k=p+l Since G i e D(Km) and p G i for i 1,2,3, we see from the definition of g.c.r.'s that @p(Gi) is Ko-cOnvex for i 1,2,3 [5].In view of this and the fact that p(G 1/(P.p) p(G2) p(G 3 for k--1,2,...r for k r+l,...,p for k p+l q, we conclude that Bi/A i #p(Gi) for i 1,2,3.Therefore, there exist elements p e G i q i such hat Bi/A i I/(P i -O).Now, (4.9) implies (since A + A 2 + A 3 Next, we take up the most general theorem of this paper, which we establish via application of Theorem 3.1.
THEOREM 4.4.Let M be a fully supportable subset of V and, for k 1,2,...,q, * E(k) let Pk e P and n k o Eo(N'Gk) be circular cones in E such that EPk(X,Y) _c TGk(X,y) for all (x,y) e N and for all k.If #(Q;x 1,x) is the algebra-valued generalized polar of the product Q(x) (cf. (2.1)-(2.3))with > 0 for k _< p (p < q) and m k < 0 for k > p, then 0(Q;x 1,x) e M for all linearly independent elements x, of E such that x.i e E-f q E'k'o( and Ts(Xo,yo ), where S(xo,Yo is the set as defined in Theorem 3.1.

PROOF. If
e V-M, there is a unique nontrivial scalar homomorphism L on V such that L() 0 but L(v) * 0 for all v e: M. Now, LP k Pnk (2.4) and it (x,y) _c k(x,y) _c TGk(X ,y) for all (x,y) e N and for can be easily shown that Pk all k.
In view of remark 2.1 and the discussion immediately preceding it (with the notations therein), we have L(' (Q;x ,x)) both sides using the same m k's.Applying Theorem 3.1 to the generalized polar (LQ;Xl,X) of the product LQ of the polynomials LPk, we see that (LQ;x l,x) # 0 for all linearly independent elements x,x of E as claimed.
Consequently, the relations (4.11) implies that (Q;x l,x) # for all x,x as claimed.Finally, the arbitrary nature of (in V-M) completes the proof. q The following version of Theorem 4.4 for the case whenm k 0 and n q E(k)o k=l k=l is a result exclusively in terms of algebra-valued generalized polars with a vanishing weight.The proof is immediate as in the case of Theorem 3.2.
THEOREM 4.5.Under the notations and hypotheses of Theorem 4.4 if we assume that q q E(k) and Z m k 0 then (Q;x l,x) e M for all linearly independent x e E-Uk=l o where Eo =-Eo(N,Gq+I) i8 the cone as defined in Theorem 3.2.
Since Theorem 4.4 reduces to Theorem 3.1 on taking V K and M K {o} (c.f.Remark 4.1), it becomes the most general result of this paper.Besides, it leads to the following corollary, which combines two earlier results due to the authors [I] and [2] which includes in it (as a natural consequence) a number of other known results due to Zaheer [5], [17], Marden [3], Walsh [22], and to Bcher [4].COROLLARY 4.6.Let Eo,i Eo(N'G'i )' i=l ,2,3, be rculcm cones in E. Under the notations and hypotheses of Theorem Eo,2 for r < k_< p (4.12) E for p < k < q, o,3 then (Q;x ,x) e M for all linearly independent elements x,x of E such that 3 L 2 x e E-i=In 3 Eo,i x e E-i=lU Eo,i) u Ts'(Xo,yo ), where(x o,yo) e N o [x,x I], --I k=r+l k=p+l PROOF.If x,x are linearly independent elements of E such that x 3 Eo" i=l i and x u 3 E U Ts,(X yo ), where (x ,yo is the unique element of N Lx,x I] , q q R {p e Kl.rmkl(P -pk) I: mk)l(P -Y/(); Pl "''P e k--I G fpr p < k <_ q, * E(k) ).
Therefore we see that /B S(x ,yo ).Consequently, x and x are O (k) and x u q E (k)) linearly independent elements of E such that x n q E o k-I k=l u Ts(Xo,Yo).By Theorem4.4, )(Q;x l,x) M, as was to be proved.q If l m k # 0, Corollary (4.6) is Theorem 4.3 of a paper due to the authors [2], k=l and if (in addition) V  K and M=K {o}, it is Theorem 3.1 in the same paper.M for all linearly independent elements x,x i=l O,i 4 of E such that x E E i--Iu Eo,i, where Eo,4 -= E o(N,G4') is the cone defined by __I G 4 (Xo,Yo) {P e ,I<I('3'P2'Pl zA"/AI; Pi e Gi (Xo'Yo)} PROOF.If x,x are linearly independent elements of E such that x u E i=l o,i' then there exist a unique element (Xo,Yo) N N [2 [X,Xl] and a unique set of scalars (with 6   BY O) such that x SXo + By and Xl YXo + 6Yo" Then (Lemma 4.2).We divide the proof into the following two cases: Case (i).18 .In this case a/6 R% u () u u 3 Gi(xo,Yo)), i=l 3 and so a/8 Ts,(Xo,Yo).Finally, Corollary 4.6 says that #(Q;Xl,X) M. as was to be proved.
Case (ii)./8 m.In the case under consideration 6 M, there exists a unique nontrivial scalar homomorphism L on V such that L() 0 but L(v) 0 for v E M.
Since the hypotheses of Theorem 4.4 are satisfied for the choice of circular cones given by (4.12), we proceed as in the proof of Theorem 4.4 and again observe that the polynomials LPk (--Pk' say) satisfy the hypotheses of Theorem 3.1 with the E (k) and the G k given by (4.12) and (4.13).The , O fact that Pk is an a.h.p, of degree n k from E to K allows us to write it in the form , n Pk (sx + tx I) k (6jk s Tjkt), k 1,2,...,q.6Jk 0 for all k.Now, proceeding exactly on J=l the lines of proof of Theorem 3.1 (except that we replace Pk by Pk and take 8 0 all along) and using the same notations, we find that there exist elements Pk Gk(Xo'Yo) such that v k ink(6 Pk-Y)/ a" Note that all the Vk s cannot vanish simultaneously (since q E (k) + B3], say.
mk.Now, there must exist elements k=p+l for i 1,2 3, and (hence) From Remark 2.1 and equation (3.4) [5] we have q L((Q;x l,x)) (LQ;Xl,X g k).ffq Pk(X) # 0, k=l k=l (4.16) where LQ is the product of a.h.p.'s Pk" Since Pk(X) 0 for all k, (4.This contradicts equation (4.14).Consequently, (4.16) holds and (Q;x l,x) # for any e V M. That is, (Q;x l,x) e M, as was to be proved.
Finally, cases (i) and (ii) complete the proof.
The following Corollaries 4.8 and 4.9 can be proved directly from Theorem 4.4, via applications of a suitably modified form of Lemmas 4.2 and 4.3, exactly in the manner in which Corollaries 4.6 and 4.7 have been derived with the help of Lemmas 4.2 and 4.3.But it would neither be necessary nor worthwhile to do so.This is because it has already been proved in earlier papers due to the authors Corollary 4.5 [I] and Corollary 4.4 [2], that Corollary 4.8 (resp.Corollary 4.9) follows from Corollary 4.6 (resp.Corollary 4.7).We, therefore, state these without proof.COROLLARY 4.8 [17].LetEo,i Eo(N,Gi ), i 1,2, be circular cones in E. Under q the notations and hypotheses of Theorem 4.4, if Z m k # 0 and if the circular cones then (Q'x ,x) e M for all lnearly independent elements x,x of E such that x! E n E mk k=l k=p+l COROLLARY 4.9 [17].Under the same notations and hypotheses as in Corollary 4.8, q except that this time k__E1 m k 0 and Eo,l Eo,2 ' we have that (Q;x l,x) e M for all Zinearly independent elements x,x of E such that x E-E u E o,l ,2 For V K and M K {o}, the above Corollaries 4.8 and 4.9 are known results due to Zaheer [5].At the end it emerges that Theorem 4.4 of this paper happens to be the most general result known thus far on (algebra-valued) generalized polars, whether having a vanishing or a nonvanishing weight, and it includes in it all the corresponding results that have been established earlier in the papers due to Marden [II], Zaheer [5], and to the authors [I] and [2].It also includes improved versions of some well- known classical results, such as: Walsh's two-circle theorems [5], Marden's general theorem [7] expressed in Corollary 3.3, and Bocher's theorem [5].To sum up: apart from the fact that all the previously known results [3], [5], [2], have been jacketted into Theorem 4.4, the present study answers in full generality the type of problem on generalized polar pursued since 1971, and, it unifies the hitherto unnecessary and separate treatments traditionally meted out to the cases of the vanishing and the nonvanishing weight.With Theorem 4.4 in view, it may be pointed out there is no scope left for further stu.dies in this subject area, except possibly when different new concepts are developed on some other lines.
Given a nucleus N of E 2 and a circular mapping G N D(K), we define the circular cone E (N,G) by o E (N G)x,y TG(X y) o )N ,Yo) O KI k mk/(P -Pk 0; gk e Gk(Xo,Yo)} =I for all (x ,yo N. o PROOF.If x,x are any linearly independent elements such that x E and [.2

P
Gp+l(Xo,Yo) CO e [ Ii mk/(O -O k O; O k e Gk(Xo,Yo)} k=o P CO [ E mk/(O -O k O; O k C k} k=o Cp+ (due to the choice of m k made above).
3.  (Ill) We remark that in Corollary 3.3, we must add the hypothesis n P C k ', k=o in order to have a nont rivial result for the rational function f(z( k mk)l( ) 0 V C k=o k=o and, hence, that Cp+ C.

COROLLARY 4 . 7 .
is again a result due to the authors [I] and which reduces to Theorem 3.1 [I] when specialized for V K and M K-{o}.q Under the notations and hypotheses of CoroIL2y 4.6 i l m k , we see that Pk(x) , 02, 0 3 must be distinct elements of K (since A I, A 2 > O, n i=l Gi(xo,Yo) and (4.14) holds).Therefore, (, 03, 02, 01) (03 01)/(03 02 A2/A I, and so a/ m R X k--! (k) Eo =-Eo (N'G k) are given byo In this case, since 0, 01 02 0 3 elements of K, the equation (4.2) holds if and only if