Generic identifiability of pairs of ternary forms

TL;DR

This paper investigates the conditions under which pairs of ternary forms can be uniquely identified, revealing that only specific classical cases allow for simultaneous identifiability, using advanced geometric techniques.

Contribution

It establishes new results on the generic identifiability of pairs of ternary forms, connecting algebraic geometry with classical problems in form identification.

Findings

Two general ternary forms are only simultaneously identifiable in classical cases.

The problem reduces to analyzing a linear system on a projective bundle.

The associated map is shown not to be birational in these cases.

Abstract

We prove that two general ternary forms are simultaneously identifiable only in the classical cases of two quadratic and a cubic and a quadratic form. We translate the problem into the study of a certain linear system on a projective bundle on the plane, and we apply techniques from projective and birational geometry to prove that the associated map is not birational.