Effective full invariant systems and sharp weight bounds for ternary forms of general order

Hilbert's 1893 memoir 'Ueber die vollen Invariantensysteme' proves that every invariant of a complex ternary form of order $n$ is a polynomial in invariants of weight at most $9n(3n+1)^8$, a bound of order roughly $n^9$ obtained non-effectively from a degree bound on algebraic relations among ten auxiliary forms in nine variables. The problem asks for the effective side that Hilbert's argument does not deliver: determine the true asymptotic growth of the minimal generating weight $w(n)$ of the invariant ring of a ternary form of order $n$; and produce an explicit full invariant system for at least one order $n\ge 5$ for which none is published.

Background

A ternary form of order $n$ is a homogeneous polynomial $F(x,y,z)$ of total degree $n$ in three variables; its coefficients form a finite-dimensional complex vector space $R_n$. The special linear group $SL_3$ (the invertible $3\times 3$ complex matrices of determinant 1) acts on $R_n$ by linear substitution of the variables. An invariant is a polynomial function $J$ on $R_n$ whose value does not change when $F$ is replaced by any substituted form $A\cdot F$ with $A\in SL_3$; equivalently $J(A\cdot F)=J(F)$. Under the larger group $GL_3$ of all invertible substitutions, an invariant satisfies $J(A\cdot F)=det(A)^{g}\,J(F)$ for a nonnegative integer $g$, called the weight (Gewicht) of the invariant; weight is the classical bookkeeping parameter for ternary invariants and is proportional to the degree of $J$ in the coefficients of $F$. A full (or complete) invariant system is a finite list of invariants $J_1,\dots,J_s$ such that every invariant is a polynomial in $J_1,\dots,J_s$; the finite existence of such systems for all $n$ follows from Hilbert's basis theorem (1890, on the theory of algebraic forms). A Nullform (null form) is a form at which every invariant without constant term takes the value 0; the set of all Nullformen is the null cone, and Hilbert's analysis of it via one-parameter degenerations of a form into a canonical null form is the historical origin of what is now called the Hilbert-Mumford stability criterion, developed in full generality in geometric invariant theory for reductive groups — the class of linear algebraic groups containing $SL_3$ for which the finiteness theorems of invariant theory hold.

The 1893 memoir establishes, in its section 14, that all invariants of a ternary form of order $n$ are polynomial functions of invariants of weight at most $9n(3n+1)^8$: the argument applies a general lemma on $h+1$ forms of common degree $m$ in $h$ variables, giving a relation of degree at most $h(m+1)^{h-1}$ — with $h=9$, the nine substitution coefficients, and $m=3n$, the ten auxiliary forms (powers of the substitution determinant $\Theta$, of degree 3, and of nine functions $B_j$ of degree $n$ in the coefficients, each homogeneous of degree $3n$) — to derive a degree bound ($\kappa\le 9n(3n+1)^8$) for the algebraic relation among those ten forms. The bound grows like $9\cdot 3^8 n^9$, and although Hilbert's method yields in principle a finite procedure for writing down a full system, the weight bound is so large that the resulting computation is far beyond reach even for small $n$ beyond the classical cases (ternary cubics, settled classically with two generating invariants, and quartics, settled from 1987 on by the thirteen Dixmier-Ohno invariants).

Modern computational invariant theory, systematized by Derksen's algorithm for computing invariant rings of reductive groups and by the Derksen-Kemper monograph, provides general algorithms with explicit but enormous degree bounds, and recent work of Derksen and Makam has produced the first exponential lower bounds for the degrees of generating invariants for closely related actions (tuples of cubic forms and tuples of tensors). What lies between the classical upper bound and the modern lower-bound techniques is the subject of this problem.

Problem Statement

Let $n\ge 2$ and let $R_n$ be the complex vector space of ternary forms of order $n$ (homogeneous polynomials of degree $n$ in three variables). Let $I_n$ be the ring of polynomial functions on $R_n$ that are unchanged by the substitution action of $SL_3$ on the variables. An invariant of weight $g$ is a polynomial function $J$ on $R_n$ with $J(A\cdot F)=det(A)^{g}J(F)$ for all invertible substitutions $A\in GL_3$; every $SL_3$-invariant is a sum of such weight-$g$ functions (decompose under the scalar matrices of $GL_3$), the weight-$0$ part consists of the constants, and the weight $g$ is proportional to the degree of $J$ in the coefficients of $F$. Since $I_n$ is finitely generated, and a finite generating set may be taken to consist of invariants of definite weight, $w(n)$ is well defined as the smallest integer $W$ such that the invariants of weight at most $W$ generate $I_n$ as a complex algebra. Hilbert proved $w(n)\le 9n(3n+1)^8$, which grows on the order of $n^9$. Solve the following. (1) Establish an upper bound $w(n)\le f(n)$ with $f(n)$ of strictly smaller asymptotic order than $9n(3n+1)^8$, or prove a lower bound $w(n)\ge h(n)$ tending to infinity at a specified rate. Identify the exact value of $w(n)$ for small $n\ge 3$ if possible. (2) For at least one order $n\ge 5$ for which no complete system is published, exhibit an explicit finite full invariant system of $I_n$ (generators with rational integer coefficients in the natural monomial basis of the coefficient variables), together with a certificate that the listed invariants generate.

References

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