Related papers: Sylvester versus Gundelfinger

In this paper we study algebraic and asymptotic properties of generating sets of algebras over orders in number fields. Let $A$ be an associative algebra over an order $R$ in an algebraic number field. We assume that $A$ is a free…

In this paper we study the algebra of graph invariants, focusing mainly on the invariants of simple graphs. All other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this algebra. In fact,…

We consider a finite dimensional $\kk G$-module $V$ of a $p$-group $G$ over a field $\kk$ of characteristic $p$. We describe a generating set for the corresponding Hilbert Ideal. In case $G$ is cyclic this yields that the algebra $\kk[V]_G$…

A minimal system of 147 homogeneous generating elements of the algebra of covariants for the binary form of degree 7 is calculated.

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf's theory of optimal subgroups and we make some improvements and simplify the theory from…

Let $P_n$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_n]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field of two elements. The Peterson hit problem is to find a minimal…

Let $V$ be a simple vertex algebra of countable dimension, $G$ be a finite automorphism group of $V$ and $\sigma$ be a central element of $G$. Assume that ${\cal S}$ is a finite set of inequivalent irreducible $\sigma$-twisted $V$-modules…

Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^n$ squares are needed. In this paper, we…

In this paper, we prove that every prime $p$ which is congruent to $4,7$ modulo $9$ is the sum of two rational cubes. This is $2/3$ of Sylvester's conjecture which has a history of nearly 150 years since 1879. In the proof, we use recent…

Let G be a Coxeter group of type A_n, B_n, D_n or I_2(N), or a complex reflection group of type G(de,e,n). Let V be its standard representation and let k be an integer greater than 2. Then G acts on S(V)^{\otimes k}. We show that the…

Consider the action of a subgroup $G$ of the permutation group on the polynomial ring $S := k[x_{1}, \ldots, x_{n}]$ via permutations. We show that if $k$ does not have characteristic two, then the following are independent of $k$: the…

Let $M$ denote the vector space of $2 \times 2$ matrices with coefficients in $\mathbb{F}_3$ and trace zero. Let $G = SL_2(\mathbb{F}_3)$. Then $G$ acts on $M$ via conjugation. Let $R =(S(M^*) \otimes \Lambda(M^*))$ be the algebra of…

For a unital ring R, a Sylvester rank function is a numerical invariant which can be described in 3 equivalent ways: on finitely presented left R-modules, or on rectangular matrices over R, or on maps between finitely generated projective…

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f…

Let $\mathbf{A}$ be a finite nilpotent algebra in a congruence modular variety with finitely many fundamental operations. If $\mathbf{A}$ is of prime power order, then it is known that there is a polynomial $p$ such that for every $n \in…

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points.…

What can be said about the subalgebras of the polynomial ring, with minimal or maximal Hilbert function? This question was discussed in a recent paper by M. Boij and A. Conca. In this paper we study the subalgebras generated in degree two…

Let G be a finite p-subgroup of GL(V), where p = char(F), and V is finite-dimensional over the field F. Let S(V) be the symmetric algebra of V, S(V)^G the subring of G-invariants, and V* the F-dual space of V. The following presents our…

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field with two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a…