The Abhyankar-Sathaye Problem asks whether any biregular embedding φ : C →֒ C can be rectified, that is, whether there exists an automorphism α ∈ AutC such that α ◦ φ is a linear embedding. Here we study this problem for the embeddings φ : C3 →֒ C4 whose image X = φ(C3) is given in C4 by an equation p = f(x, y)u + g(x, y, z) = 0, where f ∈ C[x, y]\{0} and g ∈ C[x, y, z]. Under certain additional ...

For an ideal I in a polynomial ring over a field, a monomial support of I is the set of monomials that appear as terms in a set of minimal generators of I. Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projectiv...

Let Fq be a finite field and consider the polynomial ring Fq [X]. Let Q ∈ Fq [X]. A function f : Fq [X] → G, where G is a group, is called strongly Q-additive, if f(AQ + B) = f(A) + f(B) holds for all polynomials A,B ∈ Fq [X] with degB < degQ. We estimate Weyl Sums in Fq [X] restricted by Q-additive functions. In particular, for a certain character E we study sums of the form X P E(h(P )), wher...

The Dickson Algebra on q-variables is the algebra of invariants of the action of the mod-2 general linear group on a polynomial algebra in q-variables. We study the structure of certain ideals in this algebra as a module over the Steenrod Algebra A, and develop methods to determine which elements are hit by Steenrod operations. This allows us to display a very small set of A-generators for thes...

Over a field IF of any characteristic, for a commutative associative algebra A with an identity element and for the polynomial algebra IF [D] of a commutative derivation subalgebra D of A, the associative and the Lie algebras of Weyl type on the same vector space A[D] = A ⊗ IF [D] are defined. It is proved that A[D], as a Lie algebra (modular its center) or as an associative algebra, is simple ...

We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.

In this paper we give a new degree bound for polynomial invariant rings of finite groups and give some applications.