arXiv subjects
Publications and source records attributed to Vered Moskowicz.
The two-dimensional Jacobian Conjecture says that a Keller map $f: (x,y) \mapsto (p,q) \in k[x,y]^2$ having an invertible Jacobian is an automorphism of $k[x,y]$. We prove that there is no Keller map with $[k(x,y): k(p,q)]$ prime.
Let $k$ be a field of characteristic zero, and let $f: k[x,y] \to k[x,y]$, $f: (x,y) \mapsto (p,q)$, be a $k$-algebra endomorphism having an invertible Jacobian. Write $p=a_ny^n+\cdots+a_1y+a_0$, where $n=deg_y(p) \in \mathbb{N}$, $a_i \in k[x]$, $0 \leq i \leq n$, $a_n \neq 0$, and $q=c_ry^r+\cdots+c_1y+c_0$, where $r=deg_y(q) \in \mathbb{N}$, $c_i \in k[x]$, $0 \leq i \leq r$, $c_r \neq 0$. Denote the set of prime numbers by $P$. Under two mild conditions, we prove that, if $\gcd(\gcd(n,deg_x(a_n)),\gcd(r,deg_x(c_r))) \in \{1,8\} \cup P \cup 2P$, then $f$ is an automorphism of $k[x,y]$. Removing (at least one of) the two mild conditions, we present two additional results. One of the additional results implies that the known form of a counterexample $(P,Q)$ to the two-dimensional Jacobian Conjecture, $l_{1,1}(P)=\epsilon x^{\alpha \mu}y^{\beta \mu}$, $l_{1,1}(Q)=\delta x^{\alpha \nu}y^{\beta \nu}$, where $\epsilon,\delta \in k^{\times}$, $1 < \alpha <\beta$, $d:=\gcd(\alpha,\beta) > 1$, $1 < \nu < \mu$, $\gcd(\mu,\nu)=1$, actually satisfies $d > 2$.
A result by C. C.-A. Cheng, J. H. Mckay and S. S.-S. Wang says the following: Suppose the Jacobian of $A$ and $B$ is invertible in $\mathbb{C}[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \in \mathbb{C}[x,y]$. Then $w \in \mathbb{C}[A]$. We show that in CMW's result it is possible to replace $\mathbb{C}$ by any field of characteristic zero, and we conjecture the following 'two-dimensional Centralizer Conjecture over $D$': Suppose the Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \in D[x,y]$, $D$ is an integral domain of characteristic zero. Then $w \in D[A]$. We show that if the famous two-dimensional Jacobian Conjecture is true, then the two-dimensional Centralizer Conjecture is true.
Let $f: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be a $\mathbb{C}$-algebra endomorphism having an invertible Jacobian. We show that for such $f$, if, in addition, the group of invertible elements of $\mathbb{C}[f(x),f(y),x][1/v] \subset \mathbb{C}(x,y)$ is contained in $\mathbb{C}(f(x),f(y))-0$, then $f$ is an automorphism. Here $v \in \mathbb{C}[f(x),f(y)]-0$ is such that $y = u/v$, with $u \in \mathbb{C}[f(x),f(y),x]-0$. Keller's theorem (in dimension two) follows immediately, since Keller's condition $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$ implies that the group of invertible elements of $\mathbb{C}[f(x),f(y),x][1/v]$ is contained in $\mathbb{C}(x,y)-0 = \mathbb{C}(f(x),f(y))-0$.
Let $F:\mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$ be a $\mathbb{C}$-algebra endomorphism that has an invertible Jacobian. We bring two ideas concerning the Jacobian Conjecture: First, we conjecture that for all $n$, the degree of the field extension $\mathbb{C}(F(x_1),\ldots,F(x_n)) \subseteq \mathbb{C}(x_1,\ldots,x_n)$ is less than or equal to $d^{n-1}$, where $d$ is the minimum of the degrees of the $F(x_i)$'s. If this conjecture is true, then the generalized Jacobian Conjecture is true. Second, we suggest to replace in some known theorems the assumption on the degrees of the $F(x_i)$'s by a similar assumption on the degrees of the minimal polynomials of the $x_i$'s over $\mathbb{C}(F(x_1)\ldots,f(x_n))$; this way we obtain some analogous results to the known ones.
The two-dimensional Jacobian Conjecture says that a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ that has an invertible Jacobian is an automorphism. We show that if a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ has an invertible Jacobian and if $v \in \mathbb{C}[F(x),F(y),x]$ is a product of prime elements of $\mathbb{C}[F(x),F(y),x]$, then $F$ is an automorphism, where $v$ is such that $y = u/v$, where $u \in \mathbb{C}[F(x),F(y),x]$.
The two dimensional Jacobian Conjecture says that a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, is invertible. We show that a morphism $f$ having an invertible Jacobian is invertible, in each of the following two special cases: The degree of $f(x)$ is $\leq 2$; The $(0,1)$-degrees or $(1,0)$-degrees of all monomials in $f(x)$ are of the same parity. In each case there is no restriction on the degree of $f(y)$ nor on the parity of the $(0,1)$-degrees or $(1,0)$-degrees of its monomials.
The famous Jacobian problem asks: Is a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, invertible? If we add the assumption that $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$, then $f$ is invertible; this result is due to O. H. Keller (1939). We suggest the following slight generalization of Keller's theorem: If $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ is a morphism having an invertible Jacobian, and if there exist $n \geq 1$, $a \in \mathbb{C}(f(x),f(y))^*$ and $b \in \mathbb{C}(f(x),f(y))$ such that $(ax +b)^n \in \mathbb{C}(f(x),f(y))$, then $f$ is invertible. A similar result holds for $\mathbb{C}[x_1,\ldots,x_m]$.
The famous Jacobian Conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ with invertible Jacobian, is invertible ($K$ is a characteristic zero field). A known result says that if $K[f(x),f(y)] \subseteq K[x,y]$ is an integral extension, then $f$ is invertible. We slightly generalize this known result to the following: If for some "good" $λ\in K$ (in a sense that will be explained) $m K[x,y] \neq K[x,y]$ for every maximal ideal $m$ of $K[f(x),f(y)][x+ λy]$, then $f$ is invertible. We also apply our ideas to the Jacobian Conjecture, without any further assumptions.
The famous Jacobian conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ having an invertible Jacobian is invertible ($K$ is a characteristic zero field). We show that if one of the following three equivalent conditions is satisfied, then $f$ is invertible: $K[f(x),f(y)][x+y]$ is normal; $K[x,y]$ is flat over $K[f(x),f(y)][x+y]$; $K[f(x),f(y)][x+y]$ is separable over $K[f(x),f(y)]$.
The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $α$ be the exchange involution on $K[x,y]$: $α(x)= y$ and $α(y)= x$. An $α$-endomorphism $f$ of $K[x,y]$ is an endomorphism of $K[x,y]$ that preserves the involution $α$: $f α= αf$. It was shown that if $f$ is an $α$-endomorphism of $K[x,y]$ having a non-zero scalar Jacobian, then $f$ is invertible. Based on this, we bring more results that imply that a given endomorphism $f$ having a non-zero scalar Jacobian and additional conditions involving involutions, is invertible.
The well-known Dixmier conjecture asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the $γ,δ$ conjecture, and show that it is equivalent to the Dixmier conjecture. Up to checking that in the group generated by automorphisms and anti-automorphisms of $A_1$ all the involutions belong to one conjugacy class, we show that every involutive endomorphism from $(A_1,γ)$ to $(A_1,δ)$ is an automorphism ($γ$ and $δ$ are two involutions on $A_1$), and given an endomorphism $f$ of $A_1$ (not necessarily an involutive endomorphism), if one of $f(X)$,$f(Y)$ is symmetric or skew-symmetric (with respect to any involution on $A_1$), then $f$ is an automorphism.
Dixmier's famous question says the following: Is every algebra endomorphism of the first Weyl algebra, $A_1(F)$, where $F$ is a zero characteristic field, an automorphism? Let $α$ be the exchange involution on $A_1(F)$: $α(x)= y$, $α(y)= x$. An $α$-endomorphism of $A_1(F)$ is an endomorphism which preserves the involution $α$. Then one may ask the following question, which may be called the "$α$-Dixmier's problem $1$" or the "starred Dixmier's problem $1$": Is every $α$-endomorphism of $A_1(F)$ an automorphism?
Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $α$ be the exchange involution on $A_1(K)$ given by $α(x)= y$ and $α(y)= x$. The Dixmier conjecture of Dixmier (1968) asks: Is every algebra endomorphism of the Weyl algebra $A_1(K)$ an automorphism? The aim of this paper is to prove that each $α$-endomorphism of $A_1(K)$ is an automorphism. Here an $α$-endomorphism of $A_1(K)$ is an endomorphism which preserves the involution $α$. We also prove an analogue result for the Jacobian conjecture in dimension 2, called $α-JC_2$.
The aim of this paper is to try to answer Herstein's question concerning simple rings with involution, namely: If $R$ is a simple ring with an involution of the first kind, with $dim_{Z(R)}R > 4$ and $\Char(Z(R))\neq 2$, is it true that $S^2=R$? We shall see that in such a ring $R$, $R=S^3$. We shall bring two possible criteria, each shows when $R=S^2$. The first criterion: There exist $x,y \in S$ such that $xy-yx \neq 0$ and $xSy \subseteq S^2$ $\Leftrightarrow$ $S^2=R$. The second criterion: There exist $x,y \in S$ such that $xy+yx \neq 0$ and $xKy \subseteq S^2$ $\Leftrightarrow$ $S^2=R$. Actually, those results are true without any restriction on the dimension of $R$ over $Z(R)$. In the special case of matrices (with the transpose involution and with the symplectic involution) over a field of characteristic not equal to 2, it is not difficult to find, for example, $x,y \in S$ such that $xy-yx \neq 0$ and for every $s \in S$, $xsy \in S^2$. Therefore, proving Herstein's remark that for matrices the answer is known to be positive. Similar results for $K^6$, $K^4$, $K+KSK$, $KS+K^2$, $SKS$ and $S^2K$ can also be found.
An almost PI algebra is a generalisation of a just infinite algebra which does not satisfy a polynomial identity. An almost PI algebra has some nice properties: It is prime, has a countable cofinal subset of ideals and when satisfying ACC(semiprimes), it has only countably many height 1 primes. Consider an affine prime Goldie non-simple non-PI $k$-algebra $R$ of GK dimension $<3$, where $k$ is an uncountable field. $R$ is an almost PI algebra. We give some possible additional conditions which make such an algebra primitive. This gives a partial answer to Small's question: Let $R$ be an affine prime Noetherian $k$-algebra of GK dimension 2, where $k$ is any field. Does it follow that $R$ is PI or primitive? We also show that the center of $R$ is a finite dimensional field extension of $k$, and if, in addition, $k$ is algebraically closed, then $R$ is stably almost PI.