The Cubic-Linear Linearization Conjecture

Gary H. Meisters

1 Keller’s Jacobian Conjecture Rephrased

In § 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 ℂ^{n}, 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 of the two vector variables x, y ∈ ℂ^{n}:

B_{A}(x, y) := 3[diag(Ax)][diag(Ay)]A.  (1)

Open question: Is there an n×n matrix A satisfying both of the following conditions?

Cond 1. The matrix B_{A}(x, x) is nilpotent for all x in ℂ^{n}. (A is Keller admissible.)

Cond 2. There are distinct vectors x and y in ℂ^{n} such that B_{A}(x, y)(x − y) = (x − y).

In [6] I called a complex matrix A satisfying both of these conditions a bad matrix.

Note 1. The matrix-valued bilinear function B_{A}(x, y) has the following properties:

(a) B_{A}(x, y) = B_{A}(y, x) for all vectors x, y;

(b) B_{A}(x, y)z = B_{A}(x, z)y for all vectors x, y, z;

(c) B_{A}(x, x) is the Jacobian matrix HA^{′}(x) of the cubic-homogeneous mapping H_{A}(x) := [diag(Ax)]^{3}l, where l denotes the column [1, 1, …, 1]^{T}.

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 F_{A}(x) := x − H_{A}(x) := x − [diag(Ax)]^{2}Ax. It was proved in [5, §3.3 p. 118] that a poly map F (x) := x − H(x), with H(tx) = t^{3}H(x), is injective if and only if Cond 2 is false. This is a direct consequence of the identity

F (x) − F (y) = (i\sqrt{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 + āv and y = āu + av, where a = (1 + \sqrt{3}i)/2. Cond 2 is clearly false if B_{A}(x, y) is nilpotent for all x, y; for then all eigenvalues of B_{A}(x, y) must be zero for all x, y. Cond 1 (admissibility) is equivalent to det[FA^{′}(x)] ≡ 1, which is necessary for F_{A} to be injective. See [5, Lemma 1(c) page 112 & Eq (2.2) page 110]. Thus the above open question is just Keller’s question rephrased: An answer “yes” to our question is “no” to Keller’s.

2 Linearization Conjecture for Cubic-Linear Maps

The examples given in [7] suggest that perhaps The Cubic-Linear Linearization Conjecture is true: Namely, to each cubic-linear mapping

F_{A}(x) := x − H_{A}(x) := x − [diag(Ax)]^{2}Ax = x − (1/3)B_{A}(x, x)x,  (2)

satisfying Condition 1, there corresponds a (normalized) 1-parameter family h_{s} of homeomorphisms x ↦ h_{s}(x) ≡ h(s, x) of ℂ^{n} which satisfies the Schröder Equation

h(s, s F_{A}(x)) = s h(s, x), with h(s, 0) = 0 and ∂_{x}h(s, 0) = I,  (3)

for all x in ℂ^{n} and for all complex numbers s not on the unit circle; and which therefore conjugates sF_{A}(x) to sx. In fact, each h_{s} found in [7] is actually a polyomorphism^{1} of ℂ^{n} for all but finitely many complex numbers s on the unit circle. Perhaps each Schröder function h_{s}(x), defined by (3) as a formal power series in x for each |s| ≠ 1, is at least a holomorphic automorphism of ℂ^{n}. Furthermore, it seems that one always has h_{0}(x) = F_{A}(x) and h_{∞}(x) = x for all x in ℂ^{n}.

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(tx) = t^{3}H(x). Arno collected the $100 by showing that h(s, x), defined by (3) for F (x) = (x_{1} + p(x)x_{4}, x_{2} − p(x)x_{3}, x_{3} + x_{4}^{3}, x_{4}) where p(x) = x_{3}x_{1}+x_{4}x_{2}, 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. Białynicki-Birula & M. Rosenlicht. Injective morphisms of real algebraic varieties. Proc. A.M.S. 13 (1962) 200–203.

[2] L. M. Drużkowski. 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 July 1, 1996

1polyomorphism is short for polynomial automorphism.