What is currently feasible in invariant theory for binary forms?

Question

When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system above degree 6. He discussed this limitation that year in Uber das Formensystem binaerer Formen (B.G. Tuebner, Leipzig).

So far as I can tell (by reading Kung, Sturmfels, Derksen, and Eisenbud and some correspondence with them) advances in theoretical and computer algebra have not really changed that. The complexity of calculation rises so quickly with degree that the limit of degree 6 has not been passed by much if at all. But perhaps my information is incomplete or out of date.

Is it now possible to calculate specific complete systems of invariants in higher degrees? What is the state of that?

Answers to the question Algorithms in Invariant Theory give relevant references but they do not give any clear answer. One leads to an arXiv article which describes one computer package this way.

The package calculate the set of irreducible invariants up to degree min(18, βd), but in all known computable cases this set coincides with a minimal generating set, see, for example, Brouwer’s webpage http://www.win.tue.nl/~aeb/math/invar/invarm.html

That refers to the degree of the invariants, not the degree of the form they are invariant for. I did not find a description there of which cases are computable. And that link no longer works.

Answer 1

Let $F$ be a binary form of degree $d$, namely, a homogeneous polynomial of the form $$ F(\mathbb{x})=\sum_{i=0}^{d}\left(\begin{array}{c}d\\ i \end{array}\right)f_i\ x_1^{d-i}x_2^i $$ where $\mathbb{x}$ denotes the pair of variables $(x_1,x_2)$. For $g=(g_{ij})_{1\le i,j\le 2}$ in $GL_2$, define the corresponding left action on the variables by $g\mathbb{x}=(g_{11}x_1+g_{12}x_2, g_{21}x_1+g_{22}x_2)$. This gives an action on binary forms via $$ (gF)(\mathbb{x}):=F(g^{-1}\mathbb{x})\ . $$ Now consider $C(F,\mathbb{x})=C(f_0,\ldots,f_d;x_1,x_2)$ a polynomial in these $d+3$ variables. It is classically called a covariant of the (generic) binary form $F$ if it satisfies $$ C(gF,g\mathbb{x})=C(F,\mathbb{x}) $$ for all matrices $g$ in $SL_2$. Such polynomials form a ring ${\rm Cov}_d$. It has a subring ${\rm Inv}_d$ made of polynomials in the coefficients of the form $f_0,\ldots,f_d$ only. This is the ring of invariants. It is well known that invariants do not separate orbits. However, covariants separate orbits. It is also obvious that if one knows a minimal system of generators for ${\rm Cov}_d$ then it will contain (as the degree zero in $\mathbb{x}$ subset) a minimal system for ${\rm Inv}_d$.

The minimal systems for the rings ${\rm Cov}_5$ and ${\rm Cov}_6$ were determined by Gordan in his 1868 article (and not in 1875). Then von Gall determined ${\rm Cov}_8$ around 1880 and later the harder case ${\rm Cov}_7$ in 1888.

In 1967, Shioda rederived a minimal system for ${\rm Inv}_8$ and also found all the syzygies among these generators. von Gall's system for the septimic was generating but not minimal. Six elements in his list were in fact reducible. The determination of a truly minimal system of 147 covariants for ${\rm Cov}_7$ is due to Holger Cröni (2002 Ph.D. thesis) and Bedratyuk in J. symb. Comp. 2009. In 2010, Brouwer and Popoviciu obtained the minimal systems of generators for ${\rm Inv}_9$ (92 invariants) and ${\rm Inv}_{10}$ (106 invariants).

Only very recently, Lercier and Olive managed to go beyond von Gall's 1888 results and determined the minimal systems of generators for ${\rm Cov}_9$ (476 covariants) and ${\rm Cov}_{10}$ (510 covariants).

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Addendum: Recently, for $d$ divisible by four, I produced an explicit list of invariants of degrees $2,3,\ldots,\frac{d}{2}+1$ and proved that they algebraically independent. See my article "An algebraic independence result related to a conjecture of Dixmier on binary form invariants".

Answer 2

The link already works and you may use the package. Also, the upper bound 18 for degree is not principal and the package can calculate invariants and for degree >18 after small correction.