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

The Peterson hit problem in algebraic topology is to explicitly determine the dimension of the quotient space $Q\mathcal P_k = \mathbb F_2\otimes_{\mathcal A}\mathcal P_k$ in positive degrees, where $\mathcal{P}_k$ denotes the polynomial…

It is a widely open problem to determine which monomials in the n-variable polynomial ring $K[x_1,...,x_n]$ over a field $K$ have the Gotzmann property, i.e. induce a Borel-stable Gotzmann monomial ideal. Since 2007, only the case $n \le 3$…

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…

We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients…

This paper investigates Singer's conjecture by examining the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes h}$ for specific degrees and values of $h$. Utilizing hit problem techniques, we extend previous work by Mothebe et al. and…

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve…

Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with…

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m = 5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over $\F_2[x]$ cannot all be of…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $f(x)\in\mathcal{O}_K[x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of $f(x)$ for the…

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we…

Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to find all the…

The invariant theory of finite groups can connect the coding theory to the number theory. In this paper, under this conformity, we obtain the minimal generators of the rings of E-polynomials constructed from the groups related to…

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case.…

Let K be a field and denote by K[t], the polynomial ring with coefficients in K. Set A = K[f1,. .. , fs], with f1,. .. , fs $\in$ K[t]. We give a procedure to calculate the monoid of degrees of the K algebra M = F1A + $\times$ $\times$…

Let R be the quotient of a polynomial ring over a field k by an ideal generated by monomials. We derive a formula for the multigraded Poincare' series of R, i.e., the generating function for the ranks of the modules in a minimal multigraded…

Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$, with the degree of each $x_i$ being 1, regarded as a module over the mod-2 Steenrod algebra $\mathcal A$, and let $GL_k$ be the general linear group over the…

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…

The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb F}_q$ the set $S(n)$ is not a minimal…

In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring…