[Paper Review] On Invariant Theory

Valeri V. Dolotin|ArXiv.org|Dec 14, 1995

Mathematics and Applications14 citations

TL;DR

This paper introduces a novel method for computing invariants of $n$-ary forms using discriminants of multilinear forms derived from their partial derivatives—generalizing classical discriminants and resultants. By defining hyperpolarization and hyperhessians, the authors show that these discriminants yield $GL_n$-invariant polynomials, enabling systematic generation of invariants via multidimensional determinants, with explicit formulas demonstrated for binary quartics.

ABSTRACT

Here we develop a technique of computing the invariants of $n-$ary forms and systems of forms using the discriminants of corresponding multilinear forms built of their partial derivatives, which should be cosidered as analogues of classical discriminants and resultants for binary forms.

Motivation & Objective

• To develop a systematic technique for computing invariants of $n$-ary forms using discriminants of multilinear forms.
• To generalize classical invariants like discriminants and resultants to higher-order and multilinear settings.
• To establish a framework where hyperpolarization and hyperhessians produce $GL_n$-invariant polynomials.
• To provide an algorithmic pathway for generating the full ring of invariants via multidimensional determinants.
• To demonstrate the method's effectiveness through explicit computations on binary quartic forms.

Proposed method

• Define the $K$-polarization of a homogeneous polynomial $f$ of degree $k$ as a $d$-linear form obtained by taking mixed partial derivatives of order $k_i$ for each index in $K = (k_1, \dots, k_d)$.
• Introduce the $(k_1, \dots, k_d)$-hessian as the discriminant of the $K$-polarization form, denoted $\mathcal{H}^{(k_1\dots k_d)}(f)$, which yields $GL_n$-invariant polynomials.
• Use the discriminant of a $d$-linear form (computed via algorithm from reference [1]) as the core computational engine for generating invariants.
• Apply the construction to systems of forms by forming multilinear forms from partial derivatives of multiple polynomials, leading to generalized resultants.
• Define $k$-Gramm complexes via skew-symmetric contractions of tensor powers of the cotangent bundle, yielding sequences of measures on manifolds.
• Establish a deformation of the de Rham complex via $k$-Gramm complexes, with $k = d!$ giving infinite-order differential and $k = d!/2$ recovering the de Rham complex.

Experimental results

Research questions

• RQ1 How can invariants of $n$-ary forms be systematically generated from multilinear forms derived from partial derivatives?
• RQ2 What is the relationship between the discriminant of a $K$-polarized multilinear form and the classical invariants of the original form?
• RQ3 Can higher-order generalizations of the Hessian and resultant be defined and computed algorithmically?
• RQ4 How do $k$-Gramm complexes relate to known complexes like the de Rham complex?
• RQ5 What is the structure of the ring of invariants for binary quartic forms, and can it be recovered via hyperhessians?

Key findings

• The $(1,1,1)$-hessian of a binary quartic form $f$ is a degree-4 polynomial $f^{111}$, whose discriminant satisfies $D(f^{111}) = 2^{36}3^6 \mathcal{D} \mathcal{H}^6$, where $\mathcal{D}$ is the discriminant and $\mathcal{H}$ the Hankel determinant of $f$.
• The resultant of $f$ and its $(1,1,1)$-hessian is $R(f, f^{111}) = 2^{24}3^{12} \mathcal{D}^2 \mathcal{A}^4$, where $\mathcal{A}$ is the apolar invariant.
• The hyperhessian construction generates all classical invariants of binary forms: the discriminant $\mathcal{D}$, the Hankel determinant $\mathcal{H}$, and the apolar invariant $\mathcal{A}$.
• The $K$-hessian $\mathcal{H}^{(k_1\dots k_d)}(f)$ is always a product of $GL_n$-invariants, confirming its role in generating the invariant ring.
• The $k$-Gramm complex for $k = d!/2$ recovers the de Rham complex, while for $k = d!$ it yields a complex with differential of infinite order.
• The construction provides a complete algorithmic route to compute invariants via multidimensional determinants, assuming access to discriminant computation for $d$-linear forms (as in [1]).