In this survey article, we will introduce various measures of complexity for algebraic constructions in polynomial rings over fields and show how they are often uniformly bounded by the complexity of the starting data. In problems which have a linear nature, the degree of the polynomials provide a sufficient notion of complexity. However, in the non-linear case, the more sophisticated measure o...

We describe a software package facilitating computations with symmetric functions, with an emphasis on the representation theory of general linear and symmetric groups. As an application, we implement a heuristic method for approximating equivariant resolutions of modules over polynomial rings with an action of a product of a combination of general linear and symmetric groups.

In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let S be the Laurent polynomial ring k[x1, . . . , xr, x ±1 r+1, . . . , x ±1 n ], k algebraicaly closed field of characteristic 0. We define the multigrading of S by an arbitrary finitely generated abelian group A. We construct a set of fans compatible with the multi...

Let k be a field of finite characteristic p, and G a finite group acting on the left on a finite dimensional k-vector space V . Then the dual vector space V ∗ is naturally a right kG-module, and the symmetric algebra of the dual, R := Sym(V ∗), is a polynomial ring over k on which G acts naturally by graded algebra automorphisms, and if k is algebraically closed can be regarded as the space k[V...

Last week, Ari taught you about one kind of “simple” (in the nontechnical sense) ring, specifically semisimple rings. These have the property that every module splits as a direct sum of simple modules (in the technical sense). This week, we’ll look at a rather different kind of ring, namely a principal ideal domain, or PID. These rings, like semisimple rings, have the property that every (finit...

Skew polynomial rings have invited attention of mathematicians and various properties of these rings have been discussed. The nature of ideals (in particular prime ideals, minimal prime ideals, associated prime ideals), primary decomposition and Krull dimension have been investigated in certain cases. In this article, we introduce a notion of primary decomposition of a noncommutative ring. We s...

We prove an effective Weierstrass Division Theorem for algebraic restricted power series with p-adic coefficients. The complexity of such power series is measured using a certain height function on the algebraic closure of the field of rational functions over Q. The paper includes a construction of this height function, following an idea of Kani. We apply the effective Weierstrass Division Theo...

All rings and algebras considered in this paper are commutative with identity elements and, unless otherwise specified, are to be assumed to be non-trivial. All ringhomomorphisms are unital. Let k be a field. We denote the class of commutative k−algebras with finite transcendence degree over k by C. Also, we shall use t.d.(A) to denote the transcendence degree of a k−algebra A over k, A[n] to d...