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) with counterexamples to the Jacobian Conjecture. Moreover every such component has dimension at least $n^2-1.$ In particular if the set $X(n,d)$ is irreducible, and $n\ge 3, d\ge 6$, then a generic element of this set is a counterexample to the Jacobian Conjecture.
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 counterex...
The Jacobian Conjecture is a known unsolved problem and it is the problem number 16 of the list''Mathematical Problems for the Next Century'', made by Stephen Smale, in 1998. The problem asks whether or not the Jacobian matrix of a polynomial mapping $F:\mathbb{C}^n\to\mathbb{C}^n$ at every point being invertible impli...
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 $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the vo...
In this paper, we study supersingularity of modular Jacobians $J_0(N)$ and abelian varieties $A_f$ of $\mathrm{GL}_{2}$-type from both the vertical perspective (where the prime $p$ is fixed, and the level $N$ varies) and the horizontal perspective (where the newform $f$ is fixed, and the prime $p$ varies). Vertically,...
Aarya J. Kumar, Sargam Mondal, Erick Ross et al.· 0 citations
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...
Wen Chang, Yu-Jie Luo· 2 citations
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