The Cubic-Linear Linearization Conjecture

The Cubic-Linear Linearization Conjecture

Gary H. Meisters

July 1, 1996

1. Keller's Jacobian Conjecture Rephrased

In Section 2 below I state a question, still open, I raised at Curaçao [7] in July 1994. But first some terminology and background. For vectors x in Cn, let diag(x) denote the diagonal matrix whose diagonal entries are the components of the vector x. A given n×n complex matrix A serves as the kernel matrix for the matrix-valued bilinear function

BA(x, y) := 3[diag(Ax)][diag(Ay)]A

defined at the two vector variables x, y ∈ Cn. 

An Open Question: Is there an n×n matrix A satisfying both of the following conditions?
Condition 1. The matrix BA(x,x) is nilpotent for all x in Cn.  (A is Keller admissible.)
Condition 2. There are distinct vectors x and y in Cn such that BA(x,y)(x - y) = (x - y).
In [6] I called a complex matrix A satisfying both of these conditions a bad matrix, for want of a better name.

Note 1. The matrix-valued bilinear function BA(x, y) has the following properties, which are easily verified:

  1. BA(x, y) = BA(y, x) for all vectors x and y.
  2. BA(x, y) z = BA(x, z) y for all vectors x, y and z.
  3. BA(x, x) is the Jacobian matrix H ′A(x) of the cubic-homogeneous mapping HA(x) := [diag(Ax)]3l,
    where l denotes the column vector [1,1, . . . ,1]T. Notice that [diag(Ax)]3l is the same as [diag(Ax)]2Ax.
Note 2. Ott-Heinrich Keller [4] essentially asked the question:
If det [F ′ (x)] ≡ 1 for a polynomial mapping F, is then F bijective with polynomial inverse? It suffices [1 & 8] to prove injectivity. It even suffices [2] to prove injectivity for the special cubic-linear maps

FA(x) := x - HA(x) := x - [diag(Ax)]2Ax.

It was proved in [5, §3.3, p.118] that a polynomial map F (x) := x - H(x), with H(t x) = t3H(x), is injective if and only if Condition 2 is false. This is a direct consequence of the identity

F(x) - F(y) = (i√3) [ I - B(u, v) ] (u - v)

which is the mean-value formula

F(x) - F(y) = [ I - (1/3) B(x, x) + B(x, y) + B(y, y) ] (x - y)

with the transformation x = au + áv and y = áu + av, where a = (1 + i√3)/2 and á denotes the complex conjugate of a.
Condition 2 is clearly false if BA(x, y) is nilpotent for all x, y; for then all eigenvalues of BA(x, y) are zero for all x, y.
Condition 1 (admissibility) is equivalent to det [F ′A(x)] ≡ 1, which is necessary for FA to be injective. See [5; Lemma 1(c) p.112 & Eq. (2.2) p.110]. Thus the above open question is just Keller's question rephrased: An answer ``yes'' to our question is ``no'' to Keller's.

2. The Linearization Conjecture for Cubic-Linear Maps

The examples presented in [7] suggest that perhaps The Cubic-Linear Linearization Conjecture is true: Namely,
to each cubic-linear mapping
FA(x) := x - HA(x) := x - [diag(Ax)]2Ax = x - (1/3)BA(x,x )x,
satisfying Condition 1, there corresponds a (normalized) 1-parameter family of homeomorphisms x → hs(x) ≡ h(s, x) of C n which satisfies the Schröder Equation

h(s, sFA(x)) = sh(s,x), with h(s,0) = 0 and ∂xh(s,0) = I,
for all x ∈ Cn and for all complex numbers s with |s| ≠ 1; and so hs conjugates sFA to sI.

In fact, each hs found in [7] is actually a polyomorphism (short for polynomial automorphism) of C n for all but finitely many complex numbers s on the unit circle. Perhaps each Schröder function hs(x), defined by Schröder's Eq. as a formal power series in x for each |s| ≠ 1, is at least a holomorphic automorphism of C n. Furthermore, h0(x) = FA(x) and h&inf;(x) = x, ∀ x ∈ Cn. If this conjecture is true, then Keller's Jacobian Conjecture is true. About two months after the July 1994 Curaçao Conference, Arno van den Essen produced a counterexample [3] to the stronger conjecture (for which I had offered $100 at Curaçao) that the above statement might be true for the even wider class of cubic- homogeneous polynomial maps F(x) = x - H(x) where H(t x) = t3 H(x). Arno collected the $100 by showing that h(s,x), defined by the Schröder Eq. for F(x) = (x1 + p(x)x4, x2 - p(x)x3, x3 + x43, x4) where p(x) = x3x1 + x4x2, can not be polynomial! Even if this Scalar-s Cubic-Linear Linearization Conjecture is not true, there is still the more general (weaker) possibility that the Matrix-S Cubic-Linear Linearization Conjecture is true, where the scalar s is replaced by an invertible matrix S with appropriate eigenvalues. At least we would like to know: For which matrices A does there exist such a matrix S ?

REFERENCES

  1. A. Bialynicki-Birula & M. Rosenlicht. Injective morphisms of real algebraic varieties. Proc. A. M. S. 13 (1962) 200–203.
  2. L. M. Druz·kowski. An Effective Approach to Keller's Jacobian Conjecture. Math. Ann. 264 (1983) 303–313.
  3. A. R. P. van den Essen. A Counterexample to a Conjecture of Meisters. In Automorphisms of Affine Spaces, van den Essen, editor, Kluwer Academic Publishers 1995; pages 231–233, ISBN 0–7923–3523–6. Proceedings of the Conference on Invertible Polynomial Maps held at Curaçao, The Netherlands Antilles, July 4–8, 1994.
  4. O.-H. Keller. Ganze Cremona Transformationen. Monatsh. Math. 47 (1939) 299–306; Nr. 6 & 7 in Table, p.301.
  5. G. H. Meisters. Inverting Polynomial Maps of N-Space by Solving Differential Equations. In Delay and Differential Equations; Fink, Miller & Kliemann, editors, World Scientific Pub. 1992; pages 107–166. ISBN 981–02–0891-X. Proceedings in Honor of George Seifert on his Retirement, Ames, Iowa, October 18–19, 1991.
  6. G. H. Meisters. Wanted: A Bad Matrix. Amer. Math. Monthly 102(6) (June-July 1995) 546–550.
  7. G. H. Meisters. Polyomorphisms Conjugate to Dilations. In Automorphisms of Affine Spaces, van den Essen, editor, Kluwer Academic Publishers 1995; pages 67–87. ISBN,0–7923–3523–6. Proceedings of the Conference on Invertible Polynomial Maps held at Curaçao, The Netherlands Antilles, July 4–8, 1994.
  8. W. Rudin. Injective Polynomial Maps Are Automorphisms. Amer. Math. Monthly 102(6) (June-July 1995) 540–543.
Department of Mathematics and Statistics
University of Nebraska-Lincoln
Lincoln, NE 68588–0323
gmeister@math.unl.edu 
                                               Copyright © 1996 by Gary H. Meisters