Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

The problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

Degree bounds for modular covariants

Let {V,W} be representations of a cyclic group G of prime order p over a field {\Bbbk} of characteristic p. The module of covariants {\Bbbk[V,W]^{G}} is the set of G-equivariant polynomial maps {V\rightarrow W}, and is a module over {\Bbbk[V]^{G}}. We give a formula for the Noether bound {\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})}, i.e. the minimal degree d such that {\Bbbk[V,W]^{G}} is generated over {\Bbbk[V]^{G}} by elements of degree at most d.

The Symmetric Key Equation for Reed–Solomon Codes and a New Perspective on the Berlekamp–Massey Algorithm

This paper presents a new way to view the key equation for decoding Reed–Solomon codes that unites the two algorithms used in solving it—the Berlekamp–Massey algorithm and the Euclidean algorithm. A new key equation for Reed–Solomon codes is derived for simultaneous errors and erasures decoding using the symmetry between polynomials and their reciprocals as well as the symmetries between dual and primal codes. The new key equation is simpler since it involves only degree bounds rather than modular computations. We show how to solve it using the Euclidean algorithm. We then show that by reorganizing the Euclidean algorithm applied to the new key equation we obtain the Berlekamp–Massey algorithm.