Some properties of Keller maps Reporter: Dan Yan University of Chinese Academy of Science July 21, 2014

Some properties of Keller maps

This is a joint work with M. de Bondt and Tang Guoping who is my supervisor.

Jacobian Conjecture

Let K be a field of characteristic zero and K[x] := K[x1 , . . . , xn ] be the polynomial algebra in n variables. A polynomial map F : Kn → Kn is called invertible if it has an inverse which is also a polynomial map.

Jacobian Conjecture

Problem How to determine whether a polynomial map is invertible? Let G be the inverse of F . Then G ◦ F = id. Taking determinants of the Jacobian matrix, we get det JG(F (x)) · det JF (x) = 1, which implies that det JF (x) ∈ K∗ .

Jacobian Conjecture

Jacobian Conjecture(JC) Let F : Kn → Kn be a polynomial map. If det JF (x) ∈ K∗ , then F is invertible. ♠ First posed by Keller in 1939. It is also called Keller’s problem. ♠ Still open for any dimension n ≥ 2.

Some classical results of JC

Theorem (Wang 1980) Let F be a Keller map. If deg F ≤ 2, then F is invertible. Theorem (BCW 1982) It suffices to prove the Jacobian Conjecture for all cubic homogeneous polynomial maps.

Some classical results of JC Theorem (Dru˙zkowski 1983) It suffices to prove the Jacobian Conjecture for all cubic-linear maps. Let d = (d1 , . . . , dn ) and di ≥ 2 be an integer for 1 ≤ i ≤ n. A polynomial map of the form FA = (x1 + (

n X j=1

d1

a1j xj ) , . . . , xn + (

n X

anj xj )dn )

j=1

is called a non-homogeneous power-linear map, denoted by FA = x + (Ax)∗d . If d1 = d2 = · · · = dn , then it is called power-linear map.

Some classical results of JC

Theorem (M. de Bondt and A. van den Essen 2003) It suffices to prove the Jacobian Conjecture for all cubic homogeneous polynomial maps with JH symmetric, where H is homogeneous of degree 3. Theorem (Cynk, Rusek) Let V be an affine algebraic set over K and F : V → V be an injective endomorphism. Then F is an automorphism. Corollary If a Keller map F is injective, then it is invertible.

Some classical results of JC Lemma Let F = x + H be a cubic-linear map. If det JF = 1 and rank A ≤ 4, then F is invertible. Definition Let H = (H1 , . . . , Hn ) be a polynomial map over K[x]. If JH r = 0 and JH r−1 6= 0, then r is called the nilpotency index of JH. Definition The polynomial map F = x + H is called triangular if Hi ∈ K[xi+1 , . . . , xn ] for any 1 ≤ i ≤ n − 1 and Hn ∈ K. Definition The polynomial map F = x + H is called linearly triangularizable if there exist T ∈ GLn (K) such that T −1 F (T x) is triangular.

The invertibility of some Keller maps

Theorem Let F = x + H be a cubic-linear map, if Tr J((Ax)∗3 ) = 0, then rank(A) ≤ 21 (n + δ), where δ is the number diagonal elements of A which are equal to zero.

Corollary Let F = x + H be a cubic-linear map, if Qn i=1 aii 6= 0, then the Jacobian Conjecture is true for all cubic-linear maps in dimension ≤ 9.

The invertibility of some Keller maps

Theorem Let K be a field and assume F : Kn → Kn is a polynomial map such that F (p1 ) = F (p2 ) = · · · = F (pr ) for distinct collinear pi ∈ Kn . If chr K - r 6= 1 and r ≥ deg F , then det JF ∈ / K∗ .

The equivalent problem of JC Definition Let f : Kr → Kr be polynomial maps and F : Kn → Kn be nonhomogeneous power-linear maps with n > r. We say that f and F are GZ-paired (weakly GZ-paired) through the matrices B ∈ Mr,n (K) and C ∈ Mn,r (K) if (1) f (Y ) = BF (CY ) for all Y ∈ Kr , (2) BC = Ir , (3) kerB = kerJH (kerB ⊆ kerJH), where H = F − x.

The equivalent problem of JC

Theorem Let f : Kr → Kr be a (non)homogeneous polynomial map of degree (at most) d. Then there exist an n > r and a (non)homogeneous power linear map F of degree d such that f and F are GZ-paired through some matrices B ∈ Mr,n (K) and C ∈ Mn,r (K).

The equivalent problem of JC

Theorem Let F : Kn → Kn be a (non)homogeneous polynomial map of degree d and let r ≥ n − dimK (kerJH ∩ Kn ), where H = F − x. If 0 < r < n, then there exist a polynomial map f of degree d such that f and F are weakly GZ-paired through some matrices B ∈ Mr,n (K) and C ∈ Mn,r (K).

The equivalent problem of JC

Corollary Let F : Kn → Kn be a (non)homogeneous power linear map of degree d and let r = rankA. If F is of Keller type and di ≥ 2 for all i, then r < n. If r < n, then there exists a polynomial map f : Kr → Kr of degree (at most) d such that f and F are GZ-paired through some matrices B ∈ Mr,n (K) and C ∈ Mn,r (K).

The equivalent problem of JC

Problem Let F = x + H and Hi = (ai1 x1 + ai2 x2 + · · · + ain xn )di for 1 ≤ i ≤ n. If detJF = 1 and rank(A) ≤ r, then F is invertible. Theorem The above problem is equivalent to the Jacobian Conjecture in dimension r.

Some relevant conclusions Theorem Let F = x + H, where H = (H1 , H2 , . . . , Hn )t and Hi is a homogeneous polynomial of degree di for 1 ≤ i ≤ n. If det(I + DJH|a ) 6= 0 for each λ ∈ K \ {1} and every a ∈ Kn , where D=

n1 o 1 1 1 diag (λd1 − 1), (λd2 − 1), . . . , (λdn − 1) λ−1 d1 d2 dn

then F is injective on every line that pass through the origin. In particular, if d1 = d2 = · · · = dn = d ≥ 2, then detJF = 1 is equivalent to det(DJH + I) = 1, so homogeneous Keller maps are injective on lines through the origin.

Some relevant conclusions

Theorem Let F = x + H, where H = (H1 , H2 , . . . , Hn )t and Hi = (ai1 x1 + ai2 x2 + · · · + ain xn )di for 1 ≤ i ≤ n. If det(I + DJH) = 1, where D=

o n1 1 1 1 diag (λd1 − 1), (λd2 − 1), . . . , (λdn − 1) λ−1 d1 d2 dn

for any λ 6= 1, λ ∈ K and rankA ≤ 2, then F is invertible.

Some results about polynomial automorphisms

Theorem Let F = x + H be a polynomial map, such that H is homogeneous and dimK ker JH = dimK (ker JH ∩ Kn ). If JH 3 = 0, then F is invertible and F −1 = x − H(x − H). Corollary Let F = x + H be a homogeneous power linear map. If JH 3 = 0, then deg(F −1 ) ≤ (degF )2 .

Some results about polynomial automorphisms

Question Let F = x + H be a homogeneous power linear map. If F is invertible and JH k+1 = 0, then deg(F −1 ) ≤ (degF )k . Example Let x and h be given by x = (x1 , x2 , . . . , x6 )t and h = (2x2 x6 − 2x23 − x4 x5 , 2x3 x5 − x4 x6 , x5 x6 , x25 , x26 , 0)t and f = x + h.

Some results about polynomial automorphisms

Theorem Assume F = x + H : Kn → Kn is an invertible polynomial map of degree d. If kerK[X] JH ∩ Kn has dimension n − r as a K-space, then the inverse polynomial map of F has degree at most dr .

Some results about polynomial automorphisms

Corollary Assume F = x + H : Kn → Kn is an invertible polynomial map of the form F = x + H, where H is power linear of degree d. If rank JH = r, then the inverse polynomial map of F has degree at most dr .

Some results about polynomial automorphisms

Question* Let F = x + H be a polynomial map over K[x]. If F is invertible and rankJH ≤ k, then deg(F −1 ) ≤ (degF )k . Theorem If rank JH ≤ 1 or F is linearly triangularizable, then the above Question* has an affirmative answer.

The linearly triangularizability of some Keller maps Theorem Let F = x + H be a polynomial map over K and Sr be the symmetric group of order r ≥ 2. Suppose that JH is nilpotent and that JH(x(1) ) · · · JH(x(r) ) = JH(x(σ(1)) ) · · · JH(x(σ(r)) )

(1)

for all σ ∈ Sr . Then F is linearly triangularizable. Corollary Let F = x + H and Hi = (ai1 x1 + ai2 x2 + · · · + ain xn )di for 1 ≤ i ≤ n. If JH is nilpotent and symmetric, then F is linearly triangularizable.

the Structural Conjecture ([L.M. Dru˙zkowski, An effective approach to Keller’s Jacobian Conjecture, Math. Ann. 1983]) ♣ Dru˙zkowski posed the Structural Conjecture Structural Conjecture Let F (x) = x + H and H = (Ax)∗3 . If detJF = 1, then the following conditions are equivalent (JC) det(JF (x) + JF (y)) 6= 0 for every x, y ∈ Cn (∗∗) there exists bi ∈ Cn , ctj ∈ (Cn )t such that ctj bi = 0 for every i ≥ j, i, j = 1, 2, . . . , n − 1 and F has the form: P t 3 n F (x) = x + n−1 i=1 (ci x) bi , x ∈ C .

the Structural Conjecture

Example If n = 5 and
h = 0, 0, x22 x4 , x21 x3 − x22 x5 , x21 x4 . We get a cubic-linear polynomial H through the GZ-paired. Then F = x + H is a counterexample to (JC) ⇒ (∗∗).

(2)

Irreducibility properties of Keller maps Another question related to the Jacobian Conjecture is the question of whether the coordinates of the Jacobian mapping are irreducible. If the Jacobian Conjecture holds, then the answer to the above question is affirmative. Suppose F is an automorphism. Then there exists a polynomial G such that F ◦ G = G ◦ F = id. If there exists a polynomial P1 such that P1 |F1 , then F1 = P1 P2 . We have P1 (G)P2 (G) = x1 . Thus, P1 (G) = ax1 and P2 = b, where a, b ∈ K∗ and ab = 1. Therefore, F1 is irreducible.

Irreducibility properties of Keller maps

Lemma Let K be a field and assume f ∈ K[x] such that f − ax1 is homogeneous of degree d ≥ 2 for some nonzero a ∈ K. If f is reducible, then x1 | f . Theorem Let F = x + H be a Keller map over K[x] and H is homogeneous of degree d. Then F is irreducible.

Thank youœ

Some properties of Keller maps