Related papers: The hit problem for the polynomial algebra of four…

In this note, we study some properties of the filtration of the Steenrod algebra defined from the excess of admissible monomials. We give several conditions on a cocommutative graded Hopf algebra A^* which enable us to develop the theory of…

We use the representation theory of Lie algebras and computational linear algebra to obtain an explicit formula for the hyperdeterminant of a $3 \times 3 \times 2$ array: a homogeneous polynomial of degree 12 in 18 variables with 16749…

Let $\theta$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, \theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring $\Z_K$ of integers…

Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the…

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field of two elements, and let $GL_k$ be the general linear group over…

Let $K[x]$ be a polynomial algebra in a variable $x$ over a commutative $\Q$-algebra $K$, and $\G'$ be the monoid of $K$-algebra monomorphisms of $K[x]$ of the type $\s : x\mapsto x+\l_2x^2+... +\l_nx^n$, $\l_i\in K$, $\l_n$ is a unit of…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

The purpose of this article is to give, for any commutative ring A, an explicit minimal set of generators for the ring of multisymmetric functions TS^d_A(A[x_1,...,x_r]) as an A-algebra. In characteristic zero, i.e. when A is an algebra…

We give a bound for the number of real solutions to systems of n polynomials in n variables, where the monomials appearing in different polynomials are distinct. This bound is smaller than the fewnomial bound if this structure of the…

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to…

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number…

We give a minimal system of 476 generators (resp. 510 generators) for the algebra of SL(2,C)-covariant polynomials on binary forms of degree 9 (resp. degree 10). These results were only known as conjectures so far. The computations rely on…

Let $\mm=(m_0,m_1,m_2,n)$ be an almost arithmetic sequence, i.e., a sequence of positive integers with ${\rm gcd}(m_0,m_1,m_2,n) = 1$, such that $m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they minimally generate the…

All the 62 monomial elements in the canonical basis B of the quantized enveloping algebra for type $A_4$ have been determined in [2]. According to Lusztig's idea [7], the elements in the canonical basis B consist of monomials and linear…

For a field $K$ of the second kind with respect to 2 and of characteristic different from 2, we consider the decomposition of the binomials $x^{2^n} - a$ into a product of irreducible factors over $K$ and find the explicit form of the…

We show that there are fewer than (e^2+3) 2^(k choose 2) n^k/4 non-degenerate positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This is significantly smaller than…

The submodular partitioning problem asks to minimize, over all partitions $P$ of a ground set $V$, the sum of a given submodular function $f$ over the parts of $P$. The problem has seen considerable work in approximability, as it…

This Note presents a computational algorithm for determining a basis of the cohomology of the mod 2 Steenrod algebra, $\mathrm{Ext}_{\mathcal A}^{k, k+*}(\mathbb{Z}/2, \mathbb{Z}/2)$ for $k \leq 5$, based on the well-known generators and…

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group $\operatorname{Aut} F$ for four families of binary forms. The first two families that we are interested in…