# Quantitative properties of the non-properness set of a polynomial map, a positive characteristic case

## TL;DR

This paper extends previous results on the structure of the non-properness set of polynomial maps from real and complex fields to positive characteristic fields, providing a geometric proof of an upper bound on its degree.

## Contribution

It generalizes the known properties of the non-properness set to positive characteristic fields and offers a geometric proof for the degree bound.

## Key findings

- The non-properness set is covered by polynomial curves of degree at most d.
- The degree bound for the non-properness set is generalized to positive characteristic.
- A geometric proof of the degree bound is provided.

## Abstract

Let $f:\mathbb{K}^n\rightarrow\mathbb{K}^m$ be a generically finite polynomial map of degree $d$ between affine spaces. In arXiv:1411.5011 we proved that if $\mathbb{K}$ is the field of complex or real numbers, then the set $S_f$ of points at which $f$ is not proper, is covered by polynomial curves of degree at most $d-1$. In this paper we generalize this result to positive characteristic. We provide a geometric proof of an upper bound by $d$.