arXiv subjects

Michiel de Bondt

Publications and source records attributed to Michiel de Bondt.

41 records · Page 3Linked to original sources

arXiv2012

Some remarks on the Jacobian conjecture and polynomial endomorphisms

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of degree $d$, such that $\ker \jac H$ has $n-r$ independent vectors over the base field, in particular for invertible power linear maps $x + (Ax)^{*d}$ with $\rk A = r$, the degree of the inverse of $x + H$ is at most $d^r$.

math.AG↗

arXiv2011

Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives

Let $F: C^n \rightarrow C^m$ be a polynomial map with $degF=d \geq 2$. We prove that $F$ is invertible if $m = n$ and $\sum^{d-1}_{i=1} JF(α_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = \{β+ μγ| μ\in C\} \subseteq C^n$ ($γ\ne 0$), $F|_L$ is linearly rectifiable, if and only if $\sum^{d-1}_{i=1} JF(α_i) \cdot γ\ne 0$ for all $α_i \in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d \le 3$. We also prove that if $m = n$ and $\sum^{n}_{i=1} JF(α_i)$ is invertible for all $α_i \in C^n$, then $F$ is a composition of an invertible linear map and an invertible polynomial map $X+H$ with linear part $X$, such that the subspace generated by $\{JH(α) | α\in C^n\}$ consists of nilpotent matrices.

math.AC↗

arXiv2010

An ode to Phipps' jeep convoys

The jeep problem was first solved by O. Helmer and N.J. Fine. But not much later, C.G. Phipps formulated a more general solution. He formulated a so-called convoy or caravan variant of the jeep problem and reduced the original problem to it. We shall refine the convoy idea of Phipps and subsequently view a more general jeep problem, which we solve for jeep convoys as well as for a single jeep. In the last section we solve Maddex' jeep problem.

math.HO↗

arXiv2010

Jeep variants

The jeep problem was first solved by O. Helmer and N.J. Fine. But not much later, C.G. Phipps formulated a more general solution. He formulated a so-called convoy or caravan variant of the jeep problem and reduced the original problem to it. The convoy idea of Phipps was refined in [3]. Here we will apply this refined idea to several variants of the jeep problem.

math.HO↗

arXiv2010

Computing preimages of points and curves under polynomial maps

In this paper, we give two algorithms to compute preimages of curves under polynomial endomorphisms. In particular, this gives an efficient way of computing preimages of points. Furthermore, we explain the abstract setting under which one can iteratively compute the inverse of a polynomial automorphism.

math.AG↗