Related papers: A Counterexample to Weitzenb\"ock's Theorem in Cha…

Hilbert's 14th Problem asks the following question. Given a linear representation $ \beta: G \to \operatorname{GL}(\mathbf{V}) $ of a linear algebraic group over a field $ k $ is the ring $ S_{k}(\mathbf{V}^{\ast}) $ a finitely generated $…

Let $G$ be a finite group and let $k$ be a sufficiently large finite field. Let $R(G)$ denote the character ring of $G$ (i.e. the Grothendieck ring of the category of ${\mathbb{C}}G$-modules). We study the structure and the representations…

Miyanishi proved that the ring of invariants of any $\mathbb{G}_a$ action on $\mathbb{A}^3$ is $\mathbb{A}^2$, when the field $k$ has zero characteristic. However, it is not known if this result holds when $k$ has positive characteristic.…

Let $G$ be a reductive group over a field $k$ of characteristic $\neq 2$, let ${\mathfrak g}=\Lie(G)$, let $\theta$ be an involutive automorphism of $G$ and let ${\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}$ be the associated symmetric…

Let $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m > 0$, we study the invariant rings $K[V^{ m}]^G$ for the diagonal action of $G$ on $V^m$. In characteristic zero, a theorem…

Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is…

Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack \hat G over k, the metaplectic extension of the Greenberg realization of Sp_{2n}(R). We also…

The classical theorem of Weitzenboeck states that the algebra of invariants of a single unipotent transformation $g$ in $GL_m(K)$ acting on the polynomial algebra $K[x_1,...,x_m]$ over a field $K$ of characteristic 0 is finitely generated.…

Let $k$ be a field and $G \subseteq Gl_n(k)$ be a finite group with $|G|^{-1} \in k$. Let $G$ act linearly on $A = k[X_1, \ldots, X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional…

We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of G\"obel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most…

Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of…

We construct, for every integer $N\in\mathbb{N}^*$, a structure whose Grothendieck ring is isomorphic to $(\mathbb{Z}/N\mathbb{Z})[X]$, thus proving the existence of structures with a non-zero Grothendieck ring with non-zero characteristic.…

Roberts, Freudenburg, and Daigle and Freudenburg have given the smallest counterexamples to Hilbert's fourteenth problem as rings of invariants of algebraic groups. Each is of an action of the additive group on a finite dimensional vector…

We explicitly construct a finite set of separating invariants for the basic $\Ga$-actions. These are the finite dimensional indecomposable rational linear representations of the additive group $\Ga$ of a field of characteristic zero, and…

We prove the Nonvanishing Theorem for threefolds over an algebraically closed field $k$ of characteristic $p >5$.

Let $k$ be an algebraically closed field of positive characteristic $p$. In this article, we classify representations of $\mathbb{G}_a \rtimes \mathbb{G}_m$ into ${\rm SL}(3, k)$, and thereby we classify fundamental representations of…

Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of…

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic…

Let $\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\X)$ for an ergodic action $(T_g)_{g \in \F^\omega}$ of the infinite abelian group $\F^\omega$ on a probability space…

In commutative algebra, a Weitzenb\"ock derivation is a nonzero triangular linear derivation of the polynomial algebra $K[x_1,...,x_m]$ in several variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock states…