Hilbert's 14th Problem: Finite Generation of Rings

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $k$ be a field, $V$ a finite-dimensional $k$-vector space, and $G$ a linear algebraic group over $k$ with a rational linear representation on $V$. Write $k[V]=\operatorname{Sym}_k(V^*)$ for the ring of polynomial functions on $V$, identified with $k[x_1,\ldots,x_n]$ after choosing a basis. Let $k[G]$ be the coordinate ring of $G$ and let $\rho^*:k[V]\to k[G]\otimes_k k[V]$ be the pullback on polynomial functions of the action morphism $\rho:G\times V\to V$. Its invariant ring is \[ k[V]^G=\{f\in k[V]:\rho^*(f)=1\otimes f\}. \] This algebraic-group invariance condition expresses $f(gv)=f(v)$ as an identity of regular functions and distinguishes it from invariance under only the possibly finite group $G(k)$. A $k$-algebra $R$ is finitely generated if $R=k[f_1,\ldots,f_m]$ for finitely many $f_i\in R$.

Statement recorded in the supplied source.

For every field $k$, every finite-dimensional $k$-vector space $V$, and every rational linear action of a linear algebraic group $G$ on $V$, is the invariant ring $k[V]^G$ a finitely generated $k$-algebra? [S1][S2] The question recorded here is the linear-representation formulation. It has a negative answer: Nagata gave counterexamples, and Totaro's Theorem 0.1 supplies linear-representation counterexamples over every field. [S2]

Short English statement

Is the ring of invariant polynomials for every linear algebraic-group representation generated by finitely many polynomials? The historical universal claim is false.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 88 (HIL-014). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Burt Totaro, Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves, arXiv:0808.0695v1, 5 August 2008, introduction and Theorem 0.1, p. 1; section 1, p. 2, distinguishing the linear-representation question from broader variants. https://arxiv.org/pdf/0808.0695v1.