By the works of Michiel de Bondt and Arno van den Essen, and Ludwik Drużkowski, it is known that if $F=I+N$, where $I$ is the identity mapping and $N$ is cubic homogeneous polynomials in $n$ complex variables, is invertible, then the Jacobian conjecture is true for $n$ variables.

Consider $J(N)$, the jacobian of the cubic homogeneous polynomials, the works of the above authors shows that $J(N)$ is symmetric and nilpotent in complex variables. In joint work of Bass, Connell and Wright, at page 18, the corollary 2.2. It state that for a commutative ring $k$ and $F=I+N$ cubic homogeneous polynomials, if $F$ is invertible, then it implies that for all polynomials of $k^{n}$, if their jacobian is invertible, then they are invertible.

I have many questions about the statement:

1: Does "invertible of $F$" mean having polynomial inverse? And determinant equals to constant?

2: In the assumption, $k$ is just a commutative ring, with no restriction on real number or complex number, does the condition that the $J(N)$ being nilpotent need to hold for the field $\mathbb{R}$?

Edit For the first question, it got resolved by Theorem 2.1 in the above link. For the second question, what I want to know is, can the Jacobian conjecture be solved by proving every cubic homogeneous polynomials with their jacobian nilpotent in $\mathbb{R}$ has polynomial inverse?

By the works of Michiel de Bondt and Arno van den Essen, and Ludwik Drużkowski, it is known that if $F=I+N$, where $I$ is the identity mapping and $N$ is cubic homogeneous polynomials in $n$ complex variables, is invertible, then the Jacobian conjecture is true for $n$ variables.

Consider $J(N)$, the jacobian of the cubic homogeneous polynomials, the works of the above authors shows that $J(N)$ is symmetric and nilpotent in complex variables. In joint work of Bass, Connell and Wright, at page 18, the corollary 2.2. It states that for a commutative ring $k$ and $F=I+N$ cubic homogeneous polynomials, if $F$ is invertible, then it implies that for all polynomials of $k^{n}$, if their jacobian is invertible, then they are invertible.

I have many questions about the statement:

1: Does "invertible of $F$" mean having polynomial inverse? And determinant equals to constant?

2: In the assumption, $k$ is just a commutative ring, with no restriction on real number or complex number, does the condition that the $J(N)$ being nilpotent need to hold for the field $\mathbb{R}$?

Edit For the first question, it got resolved by Theorem 2.1 in the above link. For the second question, what I want to know is, can the Jacobian conjecture be solved by proving every cubic homogeneous polynomials with their jacobian nilpotent in $\mathbb{R}$ has polynomial inverse?

By the works of Michiel de Bondt and Arno van den Essen, and Ludwik Drużkowski, it is known that if $F=I+N$, where $I$ is the identity mapping and $N$ is cubic homogeneous polynomials in $n$ complex variables, is invertible, then the Jacobian conjecture is true for $n$ variables.

Consider $J(N)$, the jacobian of the cubic homogeneous polynomials, the works of the above authors shows that $J(N)$ is symmetric and nilpotent in complex variables. In joint work of Bass, Connell and Wright, at page 18, the corollary 2.2. It state that for a commutative ring $k$ and $F=I+N$ cubic homogeneous polynomials, if $F$ is invertible, then it implies that for all polynomials of $k^{n}$, if their jacobian is invertible, then they are invertible.

I have many questions about the statement:

1: Does "invertible of $F$" mean having polynomial inverse? And determinant equals to constant?

2: In the assumption, $k$ is just a commutative ring, with no restriction on real number or complex number, does the condition that the $J(N)$ being nilpotent need to hold for the field $\mathbb{R}$?

By the works of Michiel de Bondt and Arno van den Essen, and Ludwik Drużkowski, it is known that if $F=I+N$, where $I$ is the identity mapping and $N$ is cubic homogeneous polynomials in $n$ complex variables, is invertible, then the Jacobian conjecture is true for $n$ variables.

Consider $J(N)$, the jacobian of the cubic homogeneous polynomials, the works of the above authors shows that $J(N)$ is symmetric and nilpotent in complex variables. In joint work of Bass, Connell and Wright, at page 18, the corollary 2.2. It state that for a commutative ring $k$ and $F=I+N$ cubic homogeneous polynomials, if $F$ is invertible, then it implies that for all polynomials of $k^{n}$, if their jacobian is invertible, then they are invertible.

I have many questions about the statement:

1: Does "invertible of $F$" mean having polynomial inverse? And determinant equals to constant?

2: In the assumption, $k$ is just a commutative ring, with no restriction on real number or complex number, does the condition that the $J(N)$ being nilpotent need to hold for the field $\mathbb{R}$?

Edit For the first question, it got resolved by Theorem 2.1 in the above link. For the second question, what I want to know is, can the Jacobian conjecture be solved by proving every cubic homogeneous polynomials with their jacobian nilpotent in $\mathbb{R}$ has polynomial inverse?