References on invariant subring for $\mathbb{G}_a\curvearrowright\Bbbk[x_0,\cdots,x_n]$.

Question

Consider in characteristic zero. The $\mathbb{G}_a$-action on $\Bbbk[x_0,\cdots,x_n]$ is induced by a derivation $\partial:\Bbbk[x_0,\cdots,x_n]\to\Bbbk[x_0,\cdots,x_n]$, given by $$\partial(x_i)=x_{i+1},\quad 0\leq i<n\quad\textrm{and}\quad \partial(x_n)=0.$$

For example $$\Bbbk[x_0,x_1]^{\mathbb{G}_a}=\Bbbk[x_1],\quad \Bbbk[x_0,x_1,x_2]^{\mathbb{G}_a}=\Bbbk[x_1^2-2x_0x_2,x_2]. $$ For larger $n$, it is hard to compute by hand, but there seems an algorithm doing it.

I want to ask:

1. Are there literatures explicitly calculating these invariant subrings?
2. Do people name the algorithm? generating polynomials?

Answer 1

This is an unsolved problem from the classical invariant theory. The algebra $\Bbbk[x_0,x_1,x_2, \ldots, x_n]^{\mathbb{G}_a}$ is isomorphic to the algebra of covariants of a binary $n$-form , and has been computed seems up to $n=10$ or $n=12$. Historical overview, modern algorithms and the computations can be found here:

Leonid Bedratyuk, A complete minimal system of covariants for the binary form of degree 7, J. Symb. Comput. 44 (2009) 211-220.

Leonid Bedratyuk, On complete system of covariants for the binary form of degree 8, arXiv 0612113v1 (2006).

L.Bedratyuk, Kernels of derivations of polynomial rings and Kazimir elements, Ukrainian Mathematical Journal, vol.62, no.4 (2010), 495--517.

A. E. Brouwer & M. Popoviciu, The invariants of the binary nonic, J. Symb. Comput. 45 (2010) 709-720.

A. E. Brouwer & M. Popoviciu, The invariants of the binary decimic, J. Symb. Comput. 45 (2010) 837-843.

Holger Cröni, Zur Berechnung von Kovarianten von Quantiken, Dissertation, Univ. des Saarlandes, Saarbrücken, 2002.

G.Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, Encyclopedia of Mathematical Scieces

EDIT

I add the book

Andrzej Nowicki. Polynomial derivations and their rings of constants. Uniwersytet Mikolaja Kopernika Torun, 1994. ( see section 6.8)

and Maple package for calculating the kernel of linear locally nilpotent derivation (under certain assumptions)

Leonid Bedratyuk, The MAPLE package for $SL_2$-invariants and kernel of Weitzenböck derivations, arXiv:1101.0622 (2011)