[Paper Review] A computational approach to the discriminant of homogeneous polynomials

Laurent Busé, Jean-Pierre Jouanolou|arXiv (Cornell University)|Oct 17, 2012

Polynomial and algebraic computation9 references3 citations

TL;DR

This paper presents a computational, basis-invariant definition of the discriminant for homogeneous polynomials using the resultant, ensuring stability under change of basis and over general coefficient rings. It establishes that the discriminant is a prime polynomial (up to units) when the ring is a domain with 2 ≠ 0, and the square of a prime otherwise, providing a formal, unambiguous framework for algebraic geometry and number theory applications.

ABSTRACT

In this paper, the discriminant of homogeneous polynomials is studied in two particular cases: a single homogeneous polynomial and a collection of n-1 homogeneous polynomials in n variables. In these two cases, the discriminant is defined over a large class of coefficient rings by means of the resultant. Many formal properties and computational rules are provided and the geometric interpretation of the discriminant is investigated over a general coefficient ring, typically a domain.

Motivation & Objective

• To resolve foundational issues in discriminant theory, such as instability under specialization and ambiguity up to a constant factor.
• To define the discriminant unambiguously over general coefficient rings, including domains and arbitrary commutative rings.
• To provide a formal, computationally robust framework for the discriminant that is stable under change of basis.
• To generalize existing definitions by embedding the discriminant within the theory of the resultant, ensuring rigorous algebraic properties.
• To establish geometric and algebraic properties of the discriminant in the cases of a single homogeneous polynomial and $n-1$ polynomials in $n$ variables.

Proposed method

• The discriminant is defined as a specific instance of the resultant, leveraging Jouanolou’s formalism to ensure invariance under change of basis.
• For the case $c = n-1$, the discriminant is constructed as a universal equation of the discriminant locus via the resultant, with a full base change formula derived.
• For $c = 1$, the discriminant is defined via the Hessian determinant and related to the resultant, enabling a precise algebraic characterization.
• The paper uses ideal-theoretic techniques, including the theory of transposed Jacobians and Tor-functors, to analyze the structure of the discriminant ideal.
• Key algebraic tools include Cramer’s rule, regular sequences, and the notion of non-zero divisors in quotient rings to prove injectivity and isomorphism theorems.
• The construction is validated through graded ring isomorphisms and properties of the quotient ring $kC / ext{TF}_{rak{m}}(rak{D})$, ensuring consistency across different settings.

Experimental results

Research questions

• RQ1 How can the discriminant of a homogeneous polynomial be defined unambiguously over a general coefficient ring, avoiding dependence on field assumptions or constant factors?
• RQ2 What is the precise algebraic structure of the discriminant ideal in the case of $n-1$ homogeneous polynomials in $n$ variables over a domain?
• RQ3 How does the discriminant behave under base change, and can a universal formula be derived that preserves its geometric meaning?
• RQ4 What conditions ensure that the discriminant is a prime or square of a prime polynomial in the ring of coefficients?
• RQ5 Can the discriminant be rigorously connected to the resultant in a way that preserves its computational and formal properties?

Key findings

• The discriminant is defined as a specific resultant, ensuring invariance under change of basis and stability under specialization over any coefficient ring.
• Over a domain $k$, the discriminant is a prime polynomial if $2 \neq 0$ in $k$, and the square of a prime polynomial otherwise.
• The map $\bar{\varphi}$ is an isomorphism when $n$ is odd or $2$ is a nonzero divisor in $k$, establishing a precise isomorphism between the ring of invariants and the quotient ring.
• The Hessian determinant of the dehomogenized polynomial divides a certain ideal, and this leads to the conclusion that $a \in \mathrm{TF}_{\frak{m}}(\mathcal{D}^2)$, confirming ideal membership.
• The discriminant is shown to be a non-zero divisor in the quotient ring $kC / \mathrm{TF}_{\frak{m}}(\mathcal{D})$ under the stated conditions, which is essential for injectivity of the isomorphism.
• The paper proves that $\partial D / \partial \mathcal{E}_n$ is not a zero divisor in the quotient ring when $n$ is odd or $2$ is a nonzero divisor, which is crucial for the final isomorphism result.