# Questions tagged [jacobian-conjecture]

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30
questions

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193 views

### Injectivity of Keller maps

Let $M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$,
$(x,y) \mapsto (p,q)$, with $p,q \in \mathbb{C}[x,y]$
satisfying $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Such a polynomial map is ...

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184 views

### For which smooth varieties is the Jacobian conjecture known to be true?

Let $X$ be a smooth complex algebraic variety. Consider the following properties (RP) and (JC) of $X$:
(RP) Rolle Property. Let $f:X\to X$ be a morphism such that the differential $d_x f: T_x X \to ...

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137 views

### Flatness of certain $R \subseteq \mathbb{C}[x,y]$

The two-dimensional Jacobian Conjecture over $\mathbb{C}$ says the following:
Let $p,q \in \mathbb{C}[x,y]$ satisfy $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Then $\mathbb{C}[p,q]=...

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249 views

### Concerning a result of E. Formanek

A result of E. Formanek, in its two-dimensional version, says:
Let $k$ be a field of characteristic zero and let $R=k[x,y]$ be the polynomial ring in two variables.
Let $p,q \in R$ have invertible ...

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109 views

### Question about Jacobian subalgebra

Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.
Is it true that $\mathbb{C}[x, ...

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157 views

### 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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99 views

### 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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307 views

### 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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131 views

### 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 ...

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139 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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115 views

### What can be said about $\{\deg(f),\deg(g),\deg(h)\}$, such that $k[f,g,h]=k[t]$?

Let $f=f(t),g=g(t),h=h(t) \in k[t]$, $k$ is a field of characteristic zero.
Denote $\deg(f)=a$, $\deg(g)=b$, $\deg(h)=c$. Assume that $a \geq 2, b \geq 2, c \geq 2$.
Assume that $k[f,g] \neq k[t]$, $...

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144 views

### A Jacobian pair $(p,q)$ such that $\gcd(\deg(p),\deg(q))=2P$, $P \geq 5$ is prime

Let $p=p(x,y),q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely,
$p_xq_y-p_yq_x \in \mathbb{C}^*$. Denote $a:= \deg(p)$ and $b:= \deg(q)$,
where $\deg()$ denotes the total degree ($(1,1)$-...

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127 views

### Reference request concerning the generalized Jacobian Conjecture

On page 287, A. van den Essen says:
Furthermore one can show that it suffices to prove JC for all $n \geq 2$ and for all $F$'s of the form:
$F=(l_1,\ldots,l_r,x_{r+1}+M_{r+1},\ldots,x_n+M_n)$ ...

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140 views

### If $F(x_1,0,\ldots,0)=(x_1,0,\ldots,0)$, then $F$ is bijective?

Let $k$ be an algebraically closed field of characteristic zero and let $f_1,\ldots,f_n \in k[x_1,\ldots,x_n]$ have an invertible Jacobian, namely, the determinant ot their Jacobian matrix belongs to $...

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146 views

### 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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192 views

### 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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110 views

### 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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**1**answer

630 views

### 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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520 views

### An analogy between the ring of polynomials in two variables and another (commutative?) ring

One of the answers to this question says:
"In Serge Lang's Algebra, he says: "One of the most fruitful analogies in mathematics is that between the integers $\mathbb{Z}$ and the ring of polynomials $F[...

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139 views

### Rectangular Newton polygon of a Jacobian pair

Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero.
By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an ...

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52 views

### Invertibility under base change for the Weyl algebra instead of for the polynomial algebra

From Lemma 1.1.8, we obtain the following:
Assume that $R \subseteq S$ are commutative rings
and
$u: R[x,y] \to R[x,y]$ is an $R$-algebra endomorphism
that has an invertible Jacobian, namely,
$Jac(u(x)...

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82 views

### 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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486 views

### "Jacobian Conjecture" for $k[x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}]$?

Is there exist a similar conjecture to the famous Jacobian Conjecture with $\mathbb{C}[x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}]$ instead of
$\mathbb{C}[x_1,\ldots,x_n]$?
Namely, let $f$ be $\mathbb{...

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81 views

### Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?

There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$,
$k$ is a field of characteristic zero,
the Polarization Theorem, Corollary 5.5 by A. Joseph.
...

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430 views

### What are required topics I need to learn before tackling Jacobian Conjecture? [closed]

Dear MathOverFlow advisers,
I am very interested in learning about the Jacobian Conjecture, which I learned of its existence while reading articles about the machine learning's inverse learning ...

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97 views

### Showing that a crypto hash function is not permutation, possibly conditionally?

Let $f$ be some crypto hash function, say MD5 with output $n$ bits. Restrict the input to $n$ bits.
Cryptographer told me it is open problem if such restricted collision
exists, i.e. $f(x)=f(y),x \ne ...

**3**

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**1**answer

139 views

### 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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214 views

### 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 $Y = A^n$. ...

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369 views

### Weakened jacobian conjecture for entire functions

A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials.
The jacobian ...

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1k views

### Could the Jacobian conjecture be undecidable?

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 ...