# Questions tagged [jacobian-conjecture]

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### Shape of possible counterexamples to the Jacobian and Dixmier Conjectures

Let $k$ be a field of characteristic zero. It is well-known, see for example Corollary 10.2.21, that if $(x,y) \mapsto (p,q) \in k[x,y]^2$ is a counterexample to the two-dimensional Jacobian ...
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### A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$

Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper: Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra ...
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### The Jacobian Conjecture over a commutative $\mathbb{Q}$-algebra which is not an integral domain

Let $R$ be a commutative $\mathbb{Q}$-algebra which is not an integral domain, for example: $R=\frac{\mathbb{Q}[t]}{(t^2-1)}$. Let $k$ be an algebraically closed field of characteristic zero, and let ...
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### Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree). I wish to prove or find a counterexample to the following claim: If ...
132 views

### Descending chain of subalgebras of $k[x,y]$

Let $k$ be a field of characteristic zero. Let $\{R_i\}_{i \in \mathbb{N}}$ be a descending chain of $k$-subalgebras of $k[x,y]$: $k[x,y]=:R_0 \supseteq R_1 \supseteq R_2 \supseteq \ldots$, such that ...
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### Must a counterexample $f$ to the $n$-dimensional JC satisfy $\cap f^i(k[x_1,\ldots,x_n])=k$?

There is a known result concerning the two-dimensional Jacobian Conjecture which says the following: Let $k$ be a field of characteristic zero. If $f:k[x,y] \to k[x,y]$ has an invertible Jacobian and ...
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### Generalizations of 'Injectivity on one line'

The main result of J. Gwozdziewicz in this paper says the following: "Let $k$ be an algebraically closed field of characteristic zero, and let $f:k[x,y] \to k[x,y]$, $(x,y) \mapsto (p,q)$, be a $k$-...
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### Radial similarity of Newton polytopes

Let $k$ be a field of characteristic zero, and assume that $p,q \in k[x,y]$ is a Jacobian pair, namely, $p_xq_y-p_yq_x \in k^*$ (= the determinant of the Jacobi matrix $\in k^*$). It is known that ...
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### Connection between the Jacobian Conjecture and number theory conjectures

In this paper E. Formanek says: The purpose of this paper is to point out a connection between certain differential equations which have arisen in attempts to establish the two-variable Jacobian ...
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### Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?

Let $A_1:=A_1(x,y,k)$ be the first Weyl algebra over a field $k$ of characteristic zero, namely, the $k$-algebra generated by $x$ and $y$ with relation $yx-xy=1$. Let $f:(x,y) \mapsto (p,q)$ be a $k$-...
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### On transforming pair of bivariate polynomials to pair of univariate polynomials by applying polynomial map

We know that a polynomial map $f(x,y), g(x,y)$ is polynomial automorphism if there exists polynomials $p(x,y)$ and $q(x,y)$ such that $f(p,q)$=x and $g(p,q)=y$. Jacobian conjecture tries to ...
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### Jacobian Conjecture, Cubic-Keller maps

I have recently read an interesting article about the Jacobian Conjecture, in particular the reduction to the case $f(x) = x + A(x)^3$. I was wondering about codimension one divisors on $Y = A^n$. ...
Most of us know the Jacobian conjecture. Here's a version below for fixed positive integers $d$ and $n$: $J(d,n)$: If $f: C^n \rightarrow C^n$ is a polynomial map of degree $d$, and if the Jacobian ...