Questions tagged [jacobian-conjecture]

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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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Jacobian-like conjecture about the derivations of a polynomial algebra

Let $A = k[x_1,\ldots, x_n]$ be a polynomial algebra over a field of characteristic $p$. Let $Der_k(A)$ denote the Lie algebra of derivations of $A$. As we know, the Jacobian conjecture provides a ...
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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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