Let $k$ be a field of characteristic $2$. We exhibit an explicit polynomial endomorphism $F: \mathbb{A}_k^3\to\mathbb{A}_k^3$ whose Jacobian determinant is identically $1$, whose induced extension of rational function fields has degree $3$, and which is nevertheless noninjective. Since $2\neq 3$, this gives a counterexample to the usual Adjamagbo, or separable, formulation of the Jacobian conjecture in characteristic $2$. Stabilization yields analogous counterexamples in every dimension $n\geq 3$
Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to conjectures of Wehlau concerning norms. We give faithful four-dimensional representations of $C_2^3$ and $C_2^4$ over $\mathbb F_8$ for which every nonlinear orbit no...
We show that the set $A(n, d)$ of polynomial automorphisms $F : \Bbb C^n \to \Bbb C^n$ of degree at most $d$ and with $Jac(F ) = 1$ is Zariski closed. In particular every irreducible component of the set $A(n,d)$ of polynomial mappings with Jacobian $1$ is either composed with polynomial automorphisms or (generically)...
The determinant approximation conjecture states that for a residually finite group $G$ and a chain $G_i$ of finite index normal subgroups with trivial intersection, the Fuglede-Kadison determinant of every matrix over the group algebra $\mathbb{Q} G$ is the limit of the Fuglede-Kadison determinants of the matrix reduct...
Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7 + \beta x^6y$ for $\alpha^2+\beta^2 \neq 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg} \, p =7$ and $\mathrm{de...
Let $X$ be a smooth projective variety, $f:X\to X$ an int-amplified endomorphism, and $L$ any ample line bundle on $X$. We prove that, in every codimension, the total degree of prime cycles that become totally invariant under an iterate of $f$ satisfies an explicit Hilbert-function bound on their total $L$-degree. In p...
Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/...
Niels Feld· 1 citation
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