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We define the complete numerical radius norm for homomorphisms from any operator algebra into B ( H ) \mathcal B(\mathcal H) , and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K K is a complete C C -spectral set for an operator T T , then it is a complete M M -numerical radius set, where M = 1 2 ( C + C − 1 ) M=\frac 12(C+C^{-1}) . In particular, in view of Crouzeix’s theorem, there is a universal constant M M (less than 5.6) so that if P P is a matrix polynomial and T ∈ B ( H ) T \in \mathcal B(\mathcal H) , then w ( P ( T ) ) ≤ M ‖ P ‖ W ( T ) w(P(T)) \le M \|P\|_{W(T)} . When W ( T ) = D ¯ W(T) = \overline {\mathbb D} , we have M = 5 4 M = \frac 54 .

We improve the Cauchy radius of both scalar and matrix polynomials, which is an upper bound on the moduli of the zeros and eigenvalues, respectively, by using appropriate polynomial multipliers., Comment: 12 pages

We estimate the polynomial numerical indices of the spaces C ( K ) C(K) and L 1 ( μ ) L_1(\mu ) .

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A A is an ( n + 1 ) × ( n + 1 ) (n+1)\times (n+1) nonnegative matrix whose nonzero eigenvalues are: λ 0 ≥ | λ i | \lambda _0 \geq |\lambda _i| , i = 1 , … , r i=1,\ldots ,r , r ≤ n r \leq \ n , then for all x ≥ λ 0 x \geq \lambda _0 , ( ∗ ) ∏ i = 0 r ( x − λ i ) ≤ x r + 1 − λ 0 r + 1 . \begin{equation} \prod _{i=0}^{r} (x-\lambda _i) \leq x^{r+1}-\lambda _0^{r+1}.\tag *{$(\ast )$} \end{equation} To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2 ( r + 1 ) ≥ ( n + 1 ) 2(r+1)\geq (n+1) , while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 n\leq 4 and when the spectrum of A A is real. They also showed that the conjecture is asymptotically true with the dimension. Here we prove a slightly stronger inequality than in ( ∗ ) (\ast ) , from which it follows that the Boyle–Handelman conjecture is true. Actually, we do not start from the assumption that the λ i \lambda _i ’s are eigenvalues of a nonnegative matrix, but that λ 1 , … , λ r + 1 \lambda _1,\ldots , \lambda _{r+1} satisfy λ 0 ≥ | λ i | \lambda _0\geq |\lambda _i| , i = 1 , … , r i=1,\ldots , r , and the trace conditions: ( ∗ ∗ ) ∑ i = 0 r λ i k ≥ 0 , for all k ≥ 1. \begin{equation} \sum _{i=0}^{r} \lambda _i^k \geq 0, \ \mbox {for all} k \geq 1.\tag *{$(\ast \ast )$} \end{equation} A strong form of the Boyle–Handelman conjecture, conjectured in 2002 by the present authors, says that ( ∗ * ) continues to hold if the trace inequalities in ( ∗ ∗ ** ) hold only for k = 1 , … , r k=1,\ldots ,r . We further improve here on earlier results of the authors concerning this stronger form of the Boyle–Handelman conjecture.

Let S be a polynomial of degree 2n + 2, that is, positive on the real axis, and let w = 1/S on (―∞, ∞). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical Bernstein-Szegő formula for (―1,1).

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let A A be an n × n n \times n symmetric matrix with entries in the polynomial ring R [ x 1 , … , x m ] \mathbb R[x_1,\ldots ,x_m] . The result is that if A A is positive semidefinite for all substitutions ( x 1 , … , x m ) ∈ R m (x_1,\ldots ,x_m) \in \mathbb R^m , then A A can be expressed as a sum of squares of symmetric matrices with entries in R ( x 1 , … , x m ) \mathbb R(x_1,\ldots ,x_m) . Moreover, our proof is constructive and gives explicit representations modulo the scalar case.

Suppose that E → B E\to B is a vector bundle with a linear periodic map of period p p ; the map is assumed free on the outside of the 0 0 -section. A polynomial c E ( y ) c_{E}(y) , called a mod p p Chern polynomial of E E , is defined. It is analogous to the Stiefel-Whitney polynomial defined by Dold for real vector bundles with the antipodal involution. The mod p p Chern polynomial can be used to measure the size of the periodic coincidence set for fibre preserving maps of the unit sphere bundle of E E into another vector bundle.

Let p(t) be a monic polynomial. We obtain two bounds for zeros of p(t) via the Perron root and the numerical radius of the companion matrix of the polynomial.

We construct an even weight W monotone on the right half line such that the logarithmic integral of the largest log-convex minorant of W converges and the polynomials are dense in C(W). The problem of weighted polynomial approximation posed by S. Bernstein in 1924 [2] is formulated as follows. Given a weight, that is, a lower semi-continuous function W: JR -* [1, +oo] such that limjxjo xh/W(x) = 0, n > 0 consider the space C(W) consisting of all functions f continuous on JR and such that lim f (x)/W(x) = O. IxI-oo Identifying elements in C(W) that coincide on the set {x E JR: W(x) = +oo} and equipping C(W) with the norm Ilf = supxR If(x)/W(x), we get a separable normed space. The set P of all polynomials is a subset of C(W). The weighted polynomial approximation problem is to find out whether the polynomials are dense in C(W). Note that, if W,W1 are weights, W < W1, and Closc(w)P = C(W), then Clos C(W)P = C(W1). Furthermore, if W: JR -* [1, +oo] is an arbitrary function, and Wo is its lower semicontinuous regularization, Wo (x) = lim inft~x W(t), then for every continuous f, sups If(x)/W(x) I = supsR If(x)/Wo(x) 1. Thus, in fact, the lower semicontinuity condition on W is not a genuine restriction. Two (essentially different) solutions to the Bernstein problem were proposed by S. Mergelyan, N. Ahiezer-S. Bernstein and H. Pollard in 1953-1954 and by L. de Branges in 1959 [3]; for more information on the history of the Bernstein problem and related matters see [1], [9], [8], Chapter VI. Theorem A (S. Mergelyan [9]). The polynomials are dense in C(W) if and only if log W*(W x =d? J 2 + I where W*(x) = sup{fP(x)l: P Ec, IP(t)l < (ItI + 1)W(t), t E R}. Received by the editors February 20, 1999. 2000 Mathematics Subject Classification. Primary 41A10, 46E30.

We consider the set of bounded point evaluations for polynomials with respect to the L P {L^P} -norm for a measure. We give an example of a measure where the corresponding sets of bounded point evaluations vary with the exponent p. The main ingredient is the remarkable work of K. Seip on interpolating and sampling sequences for weighted Bergman spaces.

Let P n ( α , β ) ( x ) P_n^{(\alpha ,\beta )}(x) be the Jacobi polynomial of degree n. For − 1 2 ≤ α , β ≤ 1 2 - \frac {1}{2} \leq \alpha ,\beta \leq \frac {1}{2} , and 0 ≤ θ ≤ π 0 \leq \theta \leq \pi , it is proved that \[ ( sin ⁡ θ 2 ) α + 1 2 ( cos ⁡ θ 2 ) β + 1 2 | P n ( α , β ) ( cos ⁡ θ ) | ≤ Γ ( q + 1 ) Γ ( 1 2 ) ( n + q n ) N − q − 1 2 , {(\sin \frac {\theta }{2})^{\alpha + \frac {1}{2}}}{(\cos \frac {\theta }{2})^{\beta + \frac {1}{2}}}|P_n^{(\alpha ,\beta )}(\cos \theta )| \leq \frac {{\Gamma (q + 1)}}{{\Gamma (\frac {1}{2})}}\left ( {\begin {array}{*{20}{c}} {n + q} \\ n \\ \end {array} } \right ){N^{ - q - \frac {1}{2}}}, \] where q = max ( α , β ) q = \max (\alpha ,\beta ) and N = n + 1 2 ( α + β + 1 ) N = n + \frac {1}{2}(\alpha + \beta + 1) . When α = β = 0 \alpha = \beta = 0 , this reduces to a sharpened form of the well-known Bernstein inequality for the Legendre polynomial.

We prove, among other results, that the one-dimensional local domain A A is Henselian if and only if for every maximal ideal M M in the Laurent polynomial ring A [ T , T − 1 ] A[T,{T^{ - 1}}] , either M ∩ A [ T ] M \cap A[T] or M ∩ A [ T − 1 ] M \cap A[{T^{ - 1}}] is a maximal ideal. The discrete valuation ring A A is Henselian if and only if every pseudoWeierstrass polynomial in A [ T ] A[T] is Weierstrass. We apply our results to the complete intersection problem for maximal ideals in regular Laurent polynomial rings.

A maximal polynomial is a complex polynomial that has all of its roots in the unit disk, one fixed root, and all of its critical points as far as possible from a fixed point. In this paper we determine a lower bound for the number of roots and critical points of a maximal polynomial that must lie on specified circles.

For an arbitrary function f f analytic in the disk D : | z | > 1 D:\left | z \right | > 1 and continuous in D ¯ \bar D , we show that geometric convergence in D D of best L p ( 1 ≤ p ≤ ∞ ) {L_p}(1 \leq p \leq \infty ) polynomial approximants to f f on C : | z | = 1 C:\left | z \right | = 1 is assured only when p = 2 p = 2 .

Let C C be a finite field and let C m {C_m} denote the ring consisting of all m × m m \times m matrices over C C . By a polynomial, we mean a polynomial in noncommuting indeterminates with coefficients in C C . It is shown here that a subset A A of C m {C_m} is the range of a polynomial without constant term if and only if 0 ∈ A 0 \in A and u A u − 1 ⊆ A uA{u^{ - 1}} \subseteq A for all invertible elements u ∈ C m u \in {C_m} .

By means of Galois theory, we give a product formula for the minimal polynomial G of {fo, fl ... , fn} C K[xl, ... , xn] which contains n algebraically independent elements, where K is a field of characteristic zero. As an application of the product formula, we give a simple proof of Gabber's degree bound inequality for the inverse of a polynomial automorphism. 0. INTRODUCTION Let K be a field, and let {fo, ... , fn} C K[xl, ..., xn] contain n algebraically independent polynomials over K. Then there is a unique irreducible polynomial (up to a constant factor in K*) G(yo, ... , Yn) E K[y1, .. , Yn ] such that G(fo, ... , fn) = 0. We call this G the minimal polynomial of fo, ... , fn over K. It can be viewed as a natural generalization of the minimal polynomial of an algebraic element over a field K. Minimal polynomials are very useful for studying polynomial automorphisms, as well as birational maps. See, for instance, Yu [11, 12] and Li and Yu [3, 4]. In [3] and [12], two different effective algorithms for computing minimal polynomials are given, by means of Grobner bases and Generalized Characteristic Polynomials (GCP), respectively. The following theorem is well known. Theorem 0.1. Let a be algebraic over a field K and ma(x) be the minimal polynomial of a over K. Then d ma(x) = ]J(x a(')) i=1 where a0), ... , a (d) are all roots of the polynomial ma(x) in the algebraic closure of K(a) and deg(ma(x)) = d, the number of roots of ma (x) . Received by the editors December 22, 1992 and, in revised form, May 5, 1993; presented at AMS Special Session Geometry of Affine Space, Springfield, MO, March 20-21, 1992. 1991 Mathematics Subject Classification. Primary 12E05, 12F05, 12Y05, 13B05.

We give a very simple proof of the fact that the Lorenz equations and the Maxwell-Bloch equations do not have a polynomial flow. We also give an algorithm to decide if a two-dimensional vector field over R \mathbb {R} has a polynomial flow and how to compute the solutions (in case the vector field has a polynomial flow).

This paper shows that a graded polynomial algebra over F 2 {F_2} with Steenrod algebra action possesses an analog of the adjoint representation for the cohomology of the classifying space of a compact connected Lie group.

Explicit formulas are given for polynomial generators of H ∗ B S U {H_ * }BSU as specific polynomials in the canonical polynomial generators of H ∗ B U {H_ * }BU . The method is also applied to H ∗ ( B S U ; R ) {H_ * }(BSU;R) for any coefficient ring R R and to H ∗ ( B S O ; Z 2 ) {H_ * }(BSO;{Z_2}) .

An algebraic criterion is given for a power series in n n variables over a field of characteristic 0 to be equivalent to a polynomial in n − k n - k variables over the ring of power series in k k variables. For convergent power series over the reals or complexes a geometric interpretation of the criterion is established. An analogous sufficient condition is obtained for germs of smooth functions. Most of the previously known results follow easily from the criterion.

Two theorems are proved with the aid of operator and series iteration methods. Special cases appear to give new and known generating functions for the Jacobi polynomial.

Let R R be a ring satisfying the maximal condition on annihilator left ideals and σ \sigma be an automorphism of R R . We show that the Jacobson radical of the skew polynomial ring R σ [ x ] {R_\sigma }[x] is nonzero if and only if the prime radical of R σ [ x ] {R_\sigma }[x] is nonzero. Furthermore, it is so if and only if the prime radical of R R is nonzero. In general, an example is given of a commutative semisimple algebra R R and an automorphism σ \sigma such that R σ [ x ] {R_\sigma }[x] is prime but the Levitzki radical of R σ [ x ] {R_\sigma }[x] is nonzero.

It is reasonable to expect the first sign change of a real cosine polynomial to decrease when its smallest frequency is increased. Many cases in which this is true are exhibited, but it is shown that there exist (presumably unusual) cosine polynomials for which the first sign change may increase by an arbitrarily large amount.

The polynomial Pell’s equation is P 2 − ( x 2 + d ) Q 2 = 1 {P^2} - ({x^2} + d){Q^2} = 1 , where d d is an integer and the solutions P , Q P,Q must be polynomials with integer coefficients. It is proved that this equation has nonconstant solutions if and only if d = ± 1 , ± 2 d = \pm 1, \pm 2 , and in these cases all solutions are determined.

The coefficients and the discriminant of the octic period polynomial ψ 8 ( z ) {\psi _8}(z) are computed, where, for a prime p = 8 f + 1 p = 8f + 1 , ψ 8 ( z ) {\psi _8}(z) denotes the minimal polynomial over Q {\mathbf {Q}} of the period ( 1 / 8 ) ∑ n = 1 p − 1 exp ⁡ ( 2 π i n 8 / p ) (1/8)\sum \nolimits _{n = 1}^{p - 1} {\exp (2\pi i{n^8}/p)} . Also, the finite set of prime octic nonresidues ( mod p ) (\mod p) which divide integers represented by ψ 8 ( z ) {\psi _8}(z) is characterized.

Fix real numbers α ⩾ β ⩾ − 1 2 \alpha \geqslant \beta \geqslant - \tfrac {1}{2} , with α > − 1 2 \alpha > - \tfrac {1}{2} , and equip [ − 1 , 1 ] [ - 1,1] with the measure d μ ( x ) = ( 1 − x ) α ( 1 + x ) β d x d\mu (x) = {(1 - x)^\alpha }{(1 + x)^\beta }dx . For p = 4 ( α + 1 ) / ( 2 α + 3 ) p = 4(\alpha + 1)/(2\alpha + 3) there exists f ∈ L p ( μ ) f \in {L^p}(\mu ) such that f ( x ) = 0 f(x) = 0 a.e. on [ − 1 , 0 ] [ - 1,0] and the appropriate Jacobi polynomial series for f f diverges a.e. on [ − 1 , 1 ] [ - 1,1] . This implies failure of localization for spherical harmonic expansions of elements of L 2 d / ( d + 1 ) ( X ) {L^{2d/(d + 1)}}(X) , where X X is a sphere or projective space of dimension d > 1 d > 1 .

A formula is given for the first coefficient of the Conway polynomial of a link in terms of its linking numbers. A graphical interpretation of this formula is also given. Introduction. Suppose that L is an oriented link of n components in 53. Associated to L is its Conway polynomial V?(z), which must be of the form VL(z) = z-1[a0 + alz2+ ■■■+amz2m\. Let VL(z) = VL(z)/z"~1. In this paper we shall give a formula for a0 = VL(0) which depends only on the linking numbers of L. We will also give a graphical interpretation of this formula. It should be noted that the formula we give was previously shown to be true up to absolute value in [3]. The author wishes to thank Hitoshi Murakami for bringing Professor Hosakawa's paper to his attention. We shall assume a basic familiarity with the Conway polynomial and its properties. The reader is referred to [1, 2, 4, 5 and 6] for a more detailed exposition. The fact that VL(z) has the form described above can be found in [4 or 6], for example. 1. A formula for V?(0). Suppose L [Kv K2,... ,Kn) is an oriented link in S3. Let ltj = lk(AT,, Kj) if i +j and define /„ = -jLu-ii+ihjDefine the linking matrix ££, or S^(L), as JSf= (/, ). Now JSPis a symmetric matrix with each row adding to zero. Under these conditions it follows that every cofactor =S?;y of £? is the same. (Recall thatSetj = (-l)'+ydet MtJ, where Mi} is the (i, j) minor of &.) Theorem 1. Let L he an oriented link of n components in S3. Then V?(0) = 3?ij, where ^j is any cofactor of the linking matrix Jif. Proof. Let F be a Seifert surface for L. We may picture F as shown in Figure 1.1. Let {ai} be the set of generators for Ha[F) shown in the figure and define the Siefert matrix V = {vii) in the usual way. Namely, vii} = lk(a,+, af), where a,+ is obtained by lifting ai slightly off of F in the positive direction. Then if a, n a, = 0 we have vi, = vii = lk(a,, uj). If ai n Oj # 0, then {;', j) = {2k — 1,2/c) for some Received by the editors August 14, 1984 and, in revised form, December 28, 1984. 1980 Mathematics Subject Classification. Primary 57M25.

Let k k be an infinite field and B = k [ X 1 , … , X n ] B = k[{X_1}, \ldots ,{X_n}] a polynomial ring over k k . Let M M be a finitely generated module over B B . For every prime ideal P ⊂ B P \subset B let μ ( M P ) \mu ({M_P}) be the minimum number of generators of M P {M_P} , i.e., μ ( M P ) = dim ⁡ B P / P P ( M P ⊗ B P ( B P / P P ) ) \mu ({M_P}) = \dim {B_P}/{P_P}({M_P}{ \otimes _{{B_P}}}({B_P}/{P_P})) . Set η ( M ) = max { μ ( M P ) + dim ⁡ ( B / P ) | P ∈ Spec B such that M P is not free } \eta (M) = \max \{ \mu ({M_P}) + \dim (B/P)\left | {P \in \operatorname {Spec} } \right .B\;{\text {such}}\;{\text {that}}\;{M_{P\;}}{\text {is}}\;{\text {not}}\;{\text {free}}\} . Then M M can be generated by η ( M ) \eta (M) elements. This improves earlier results of A. Sathaye and N. Mohan Kumar on a conjecture of Eisenbud-Evans.

In 1939 P. Erdös conjectured that the Chebyshev polynomial T n ( x ) {T_n}(x) has a maximal arc-length in [ − 1 , 1 ] [ - 1,1] among the polynomials of degree n n which are bounded by 1 in [ − 1 , 1 ] [ - 1,1] . We prove this conjecture for every natural n n .

Let D D be a bounded Jordan domain. Define A q ( D ) {A_q}(D) , the Bers space, to be the Banach space of holomorphic functions on D D , such that ∬ D | f | λ D 2 − q d x d y \iint _D {|f|\lambda _D^{2 - q}dxdy} is finite, where λ D ( z ) {\lambda _D}(z) is the Poincaré metric for D D . It is well known that the polynomials are dense in A q ( D ) {A_q}(D) for 2 ≦ q > ∞ 2 \leqq q > \infty and we shall prove they are dense in A q ( D ) {A_q}(D) for 1 > q > 2 1 > q > 2 if the boundary of D D is rectifiable. Also some remarks are made in case the boundary of D D is not rectifiable.

Let A A be an n × n n \times n complex matrix. For 1 ≤ k ≤ n 1 \leq k \leq n we study the inclusion relation for the following polynomial sets related to the matrix A A . (a) The classical numerical range of the k k th compound of the matrix λ I − A \lambda I - A . (b) The k k th decomposable numerical range of the matrix λ I − A \lambda I - A . (c) The convex hull of the set of all monic polynomials of degree k k that divide the characteristic polynomial of A A . Moreover, we give an example showing that the set described in (a) is not convex in general. This settles a question raised by C. Johnson.

Let K K be a compact set in the complex plane having connected and regular complement, and let f f be any function continuous on K K and analytic in the interior of K K . For the polynomials p n ∗ ( f ) p_n^*(f) of respective degrees at most n n of best uniform approximation to f f on K K , we investigate the density of the sets of extreme points \[ A n ( f ) := { z ∈ K : | f ( z ) − p n ∗ ( f ) ( z ) | = | | f − p n ∗ ( f ) | | K } {A_n}(f): = \{ z \in K:|f(z) - p_n^*(f)(z)| = ||f - p_n^*(f)|{|_K}\} \] in the boundary of K K .

We prove that there exist C ∞ {C^\infty } functions φ : R t × R x → R \varphi :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}} such that although φ ( t , x ) \varphi \left ( {t,x} \right ) is a polynomial in x x for each t t in R , φ ˙ ( 0 , x ) ≡ ( ∂ φ / ∂ t ) ( 0 , x ) {\mathbf {R}},\dot \varphi \left ( {0,x} \right ) \equiv \left ( {\partial \varphi /\partial t} \right )\left ( {0,x} \right ) need not even be analytic in x x let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if φ \varphi satisfies the group-property (even locally) of flows, namely if φ ( s , φ ( t , x ) ) = φ ( s + t , x ) \varphi \left ( {s,\varphi \left ( {t,x} \right )} \right ) = \varphi \left ( {s + t,x} \right ) .

There are given, in certain cases, necessary and sufficient conditions for a truncated polynomial ring over a Z 2 {Z_2} -Poincaré algebra to again be a Poincaré algebra. Applications are a splitting theorem for Poincaré algebras and an algebraic bordism classification for real projective space bundles.

We show that nonreal zeros of a real polynomial P ( z ) P(z) sometimes induce nonreal zeros of P ′ ( z ) P’(z) .

The rank of the n × n n \times n matrix ( A − λ I ) (A - \lambda I) is n − J ( λ ) n - J(\lambda ) when λ \lambda is an eigenvalue occurring in J ( λ ) ≧ 0 J(\lambda ) \geqq 0 Jordan blocks of the Jordan normal form of A A . In our principal theorem we derive an analogous expression for the rank of ( A − λ B ) (A - \lambda B) for general, m × n m \times n , matrices. When J ( λ ) > 0 , λ J(\lambda ) > 0,\lambda is a rank-reducing number of ( A − λ I ) (A - \lambda I) . We show how the rank-reducing properties of eigenvalues can be extended to m × n m \times n matrix expressions ( A − λ B ) (A - \lambda B) . In particular we give a constructive way of deriving a polynomial P ( λ , A , B ) P(\lambda ,A,B) whose roots are the only rank-reducing numbers of ( A − λ B ) (A - \lambda B) . We name this polynomial the characteristic polynomial of A A relative to B B and justify that name.

It is the purpose of this paper to find in terms of parameters the most general matrix X which is K-commutative with respect to a given matrix A. The proofs will yield a method of rational construction for such a matrix X.

Introduction. The location of the zeros of the derivative of a polynomial has been much studied, as has the location of the zeros of the Chebys;hev polynomial. In ?1 of the present note we set forth in a direct and elementary manner the equivalence of these two problems in a suitably specialized situation. This conclusion is mentioned (for integral 'i) with an indication of the proof by Fekete and von Neumann [4],2 and the conclusion is related to a much deeper investigation due to Fekete [3]. We obtain (?9) some new results on zeros of approximating polynomials and (??2, 3, 8) on the argument of the deviation. In ?10 we consider approximation by an arbitrary linear family. Throughout the paper we study primarily approximation on a set of n points by a polynomial of degree n-2. In ??4-7 weight functions with infinities and Chebyshev rational functions are introduced.

It is a fundamental fact about simultaneous linear equations that uniqueness implies existence in the sense that if the nXn system MX = Y has, for each Y, at most one solution X, then, for each Y, the equation actually has a solution. Our purpose is to show that this situation persists for certain "nonlinear equations" as well, namely that, restricting everything to the reals, if P1, P2 are polynomials and for each Yi, Y2 the system

This proposition has been proved by the writer [I] for the case r =2. Recently Parker [2] generalized that result and Goddard [3] gave an alternate proof of it and extended his method to the product of three matrices. This latter result does not come under the theorem above. Schneider [5] proved the theorem for the case A A1=0, i