Plane Curves With Minimal Discriminant

TL;DR

This paper establishes lower bounds for the discriminant degree of separable polynomials over algebraically closed fields, characterizes those with minimal discriminant, and explores their geometric and algebraic properties, extending classical theorems.

Contribution

It provides a geometric characterization and explicit construction of polynomials with minimal discriminant, linking algebraic invariants to geometric properties and extending classical embedding theorems.

Findings

Irreducible monic polynomials with minimal discriminant are coordinate polynomials.

Explicit constructions are provided for many cases of minimal discriminant polynomials.

Partial results relate to nonmonic or reducible polynomials via their GL2(K[x])-orbit and Newton polytope constraints.

Abstract

We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f in K[x,y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a geometrical characterisation of those polynomials having minimal discriminant, and give an explicit construction of all such polynomials in many cases. In particular, we show that irreducible monic polynomials with minimal discriminant coincide with coordinate polynomials. We obtain analogous partial results for the case of nonmonic or reducible polynomials by studying their GL2(K[x])-orbit and by establishing some combinatorial constraints on their Newton polytope. Our results suggest some natural extensions of the embedding line theorem of Abhyankar-Moh and of the Nagata-Coolidge problem to the case of unicuspidal curves of P1 x P1.