An introduction to the Jacobian conjecture Damiano Fulghesu Minnesota State University Moorhead October 12, 2010 Damiano Fulghesu An introduction to the Jacobian conjecture
Origin of the conjecture. First formulated by O. Keller in 1939. Damiano Fulghesu An introduction to the Jacobian conjecture
Origin of the conjecture. First formulated by O. Keller in 1939. Hartshorne’s Exercise 3.19 (b). Damiano Fulghesu An introduction to the Jacobian conjecture
Origin of the conjecture. First formulated by O. Keller in 1939. Hartshorne’s Exercise 3.19 (b). It is one of the 18 Smale’s problems. Damiano Fulghesu An introduction to the Jacobian conjecture
Origin of the conjecture. First formulated by O. Keller in 1939. Hartshorne’s Exercise 3.19 (b). It is one of the 18 Smale’s problems. Still unproven? Damiano Fulghesu An introduction to the Jacobian conjecture
Polynomial maps and their Jacobian Base field C . Damiano Fulghesu An introduction to the Jacobian conjecture
Polynomial maps and their Jacobian Base field C . Polynomial map F = ( F 1 , . . . , F n ) : C n C n → z = ( z 1 , . . . , z n ) �→ ( F 1 ( z ) , . . . , F n ( z )) Damiano Fulghesu An introduction to the Jacobian conjecture
Polynomial maps and their Jacobian Base field C . Polynomial map F = ( F 1 , . . . , F n ) : C n C n → z = ( z 1 , . . . , z n ) �→ ( F 1 ( z ) , . . . , F n ( z )) Jacobian ∂ F 1 ∂ F 1 ∂ z 1 ( z ) . . . ∂ z n ( z ) . . ... . . J F ( z ) = . . ∂ F n ∂ F n ∂ z 1 ( z ) . . . ∂ z n ( z ) Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Theorem If F : C n → C n is invertible (and its inverse is a polynomial map) then | J F ( z ) | ∈ C ∗ (the determinant of J F ( z ) is a nonzero constant) Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Theorem If F : C n → C n is invertible (and its inverse is a polynomial map) then | J F ( z ) | ∈ C ∗ (the determinant of J F ( z ) is a nonzero constant) Proof: Let G : C n → C n be the inverse of F . Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Theorem If F : C n → C n is invertible (and its inverse is a polynomial map) then | J F ( z ) | ∈ C ∗ (the determinant of J F ( z ) is a nonzero constant) Proof: Let G : C n → C n be the inverse of F . We must have that the compositions F ◦ G and G ◦ F are the identity maps. Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Theorem If F : C n → C n is invertible (and its inverse is a polynomial map) then | J F ( z ) | ∈ C ∗ (the determinant of J F ( z ) is a nonzero constant) Proof: Let G : C n → C n be the inverse of F . We must have that the compositions F ◦ G and G ◦ F are the identity maps. By applying the chain rule , we get J G ◦ F ( z ) = J G ( F ( z )) · J F ( z ) = I n Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Theorem If F : C n → C n is invertible (and its inverse is a polynomial map) then | J F ( z ) | ∈ C ∗ (the determinant of J F ( z ) is a nonzero constant) Proof: Let G : C n → C n be the inverse of F . We must have that the compositions F ◦ G and G ◦ F are the identity maps. By applying the chain rule , we get J G ◦ F ( z ) = J G ( F ( z )) · J F ( z ) = I n In particular, for every z , the determinant | J F ( z ) | must be different from 0. Damiano Fulghesu An introduction to the Jacobian conjecture
Invertible polynomial maps Theorem If F : C n → C n is invertible (and its inverse is a polynomial map) then | J F ( z ) | ∈ C ∗ (the determinant of J F ( z ) is a nonzero constant) Proof: Let G : C n → C n be the inverse of F . We must have that the compositions F ◦ G and G ◦ F are the identity maps. By applying the chain rule , we get J G ◦ F ( z ) = J G ( F ( z )) · J F ( z ) = I n In particular, for every z , the determinant | J F ( z ) | must be different from 0. We conclude by observing that J F ( z ) is a polynomial and C is algebraically closed . Damiano Fulghesu An introduction to the Jacobian conjecture
The Jacobian conjecture Damiano Fulghesu An introduction to the Jacobian conjecture
The Jacobian conjecture Conjecture Let F : C n → C n be a polynomial map such that | J F ( z ) | ∈ C ∗ then F is invertible (and its inverse is a polynomial map). Damiano Fulghesu An introduction to the Jacobian conjecture
The Jacobian conjecture Conjecture Let F : C n → C n be a polynomial map such that | J F ( z ) | ∈ C ∗ then F is invertible (and its inverse is a polynomial map). Still open for n ≥ 2!!! Damiano Fulghesu An introduction to the Jacobian conjecture
The Jacobian conjecture Conjecture Let F : C n → C n be a polynomial map such that | J F ( z ) | ∈ C ∗ then F is invertible (and its inverse is a polynomial map). Still open for n ≥ 2!!! More generally Instead of C , we can consider any algebraically closed field k with characteristic 0. Damiano Fulghesu An introduction to the Jacobian conjecture
The case n = 1 Damiano Fulghesu An introduction to the Jacobian conjecture
The case n = 1 The conjecture is true for n = 1 Damiano Fulghesu An introduction to the Jacobian conjecture
The case n = 1 The conjecture is true for n = 1 The map is a polynomial F ( z ), such that ∂ F ∂ z ( z ) = a � = 0 Damiano Fulghesu An introduction to the Jacobian conjecture
The case n = 1 The conjecture is true for n = 1 The map is a polynomial F ( z ), such that ∂ F ∂ z ( z ) = a � = 0 Therefore Damiano Fulghesu An introduction to the Jacobian conjecture
The case n = 1 The conjecture is true for n = 1 The map is a polynomial F ( z ), such that ∂ F ∂ z ( z ) = a � = 0 Therefore F ( z ) = az + b with a � = 0 Damiano Fulghesu An introduction to the Jacobian conjecture
The case n = 1 The conjecture is true for n = 1 The map is a polynomial F ( z ), such that ∂ F ∂ z ( z ) = a � = 0 Therefore F ( z ) = az + b with a � = 0 the inverse is G ( z ) = z − b a Damiano Fulghesu An introduction to the Jacobian conjecture
An example for n = 2 Damiano Fulghesu An introduction to the Jacobian conjecture
An example for n = 2 We define F : C 2 C 2 → z 1 + ( z 1 + z 2 ) 3 , z 2 − ( z 1 + z 2 ) 3 � � ( z 1 , z 2 ) �→ Damiano Fulghesu An introduction to the Jacobian conjecture
An example for n = 2 We define F : C 2 C 2 → z 1 + ( z 1 + z 2 ) 3 , z 2 − ( z 1 + z 2 ) 3 � � ( z 1 , z 2 ) �→ We have (Exercise): Damiano Fulghesu An introduction to the Jacobian conjecture
An example for n = 2 We define F : C 2 C 2 → z 1 + ( z 1 + z 2 ) 3 , z 2 − ( z 1 + z 2 ) 3 � � ( z 1 , z 2 ) �→ We have (Exercise): the determinant of J F ( z 1 , z 2 ) is 1 Damiano Fulghesu An introduction to the Jacobian conjecture
An example for n = 2 We define F : C 2 C 2 → z 1 + ( z 1 + z 2 ) 3 , z 2 − ( z 1 + z 2 ) 3 � � ( z 1 , z 2 ) �→ We have (Exercise): the determinant of J F ( z 1 , z 2 ) is 1 the inverse of F is G : C 2 C 2 → z 1 − ( z 1 + z 2 ) 3 , z 2 + ( z 1 + z 2 ) 3 � � ( z 1 , z 2 ) �→ Damiano Fulghesu An introduction to the Jacobian conjecture
Linear Case Damiano Fulghesu An introduction to the Jacobian conjecture
Linear Case Definition deg( F ) = max i deg( F i ) Damiano Fulghesu An introduction to the Jacobian conjecture
Linear Case Definition deg( F ) = max i deg( F i ) Up to replacing F ( z ) with F ( z ) − F (0), we can assume F (0) = 0 Damiano Fulghesu An introduction to the Jacobian conjecture
Linear Case Definition deg( F ) = max i deg( F i ) Up to replacing F ( z ) with F ( z ) − F (0), we can assume F (0) = 0 and we consider the decomposition in homogenous components F = F (1) + · · · + F ( d ) Damiano Fulghesu An introduction to the Jacobian conjecture
Linear Case Definition deg( F ) = max i deg( F i ) Up to replacing F ( z ) with F ( z ) − F (0), we can assume F (0) = 0 and we consider the decomposition in homogenous components F = F (1) + · · · + F ( d ) Theorem (Linear Algebra) If deg( F ) = 1 then the Jacobian conjecture is true. Damiano Fulghesu An introduction to the Jacobian conjecture
Reduction to injectivity Damiano Fulghesu An introduction to the Jacobian conjecture
Reduction to injectivity A. Bia� lynicki-Birula, M. Rosenlicht (1962) Let F : C n → C n be a polynomial map. If F is injective, then F is surjective. Damiano Fulghesu An introduction to the Jacobian conjecture
Reduction to injectivity A. Bia� lynicki-Birula, M. Rosenlicht (1962) Let F : C n → C n be a polynomial map. If F is injective, then F is surjective. S. Cynk, K. Rusek (1991) If F : C n → C n is a bijective polynomial map, then the inverse of F is a polynomial map. Damiano Fulghesu An introduction to the Jacobian conjecture
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