Some facts about discriminants

TL;DR

This paper investigates the geometric properties of the discriminant set of univariate polynomials, focusing on when it can be viewed as a function of all but one coefficient and analyzing its singularities.

Contribution

It provides new insights into the structure and smoothness of the discriminant set for polynomials, exploring its projections and singular points.

Findings

Identifies conditions under which the discriminant set is a graph of a function

Analyzes the projection of singular points onto coefficient hyperplanes

Characterizes the smooth and singular regions of the discriminant set

Abstract

For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\cdots +a_n$ we ask the questions at which points its discriminant set can be considered as the graph of a function of all coefficients $a_j$ but one and how its subset of points, where the discriminant set is not smooth, projects on the different coordinate hyperplanes in the space of the coefficients~$a$.