arXiv subjects

Matt Olechnowicz

Publications and source records attributed to Matt Olechnowicz.

4 recordsLinked to original sources

arXiv2026

Critical loci of self-maps of projective space

Let $K$ be an algebraically closed field and let $n, d \geq 2$. We show that the critical scheme of a general endomorphism of $\mathbb{P}^n_K$ of degree $d$ is an integral hypersurface. This extends a result of Ingram--Ramadas--Silverman to arbitrary characteristic. We use basic facts about polynomial rings to show that certain carefully chosen examples have absolutely irreducible Jacobian. To handle the wild case where the characteristic of $K$ divides $d$, we introduce a polynomial that imitates the homogeneous Jacobian determinant.

math.AG↗

arXiv2024

Heights and morphisms in number fields

We give a formula with explicit error term for the number of $K$-rational points $P$ satisfying $H(f(P)) \le X$ as $X \to \infty$, where $f$ is a nonconstant morphism between projective spaces defined over a number field $K$ and $H$ is the absolute multiplicative Weil height. This yields formulae for the counting functions of $f(\mathbb{P}^m(K))$ with respect to the Weil height as well as of $\mathbb{P}^m(K)$ with respect to the Call-Silverman canonical height.

math.NT↗

arXiv2023

Dynamically improper hypersurfaces for endomorphisms of projective space

We introduce a new generalization of the notion of preperiodic hypersurface and explore some of its basic ramifications. We also prove that among nonlinear endomorphisms of projective space, those with a periodic critical point are Zariski dense. This answers a question of Ingram.

math.DS↗

arXiv2023

Distribution of preperiodic points in one-parameter families of rational maps

Let $f_t$ be a one-parameter family of rational maps defined over a number field $K$. We show that for all $t$ outside of a set of natural density zero, every $K$-rational preperiodic point of $f_t$ is the specialization of some $K(T)$-rational preperiodic point of $f$. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of $K$-rational preperiodic points of $f$, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family $f_t(z) = z^2 + t$ over the field of rational numbers.

math.NT↗