This paper studies the link between the number of critical eigenvalues and the number of delays in certain classes of delay-differential equations. There are two main results. The first states that for k purely imaginary numbers which are linearly independent over the rationals, there exists a scalar delay-differential equation depending on k fixed delays whose spectrum contains those k purely ...

We propose two local invariants for the inductive proof of the embedded resolution of purely inseparable surface singularities of order equal to the characteristic. The invariants are built on an detailed analysis of the so called kangaroo phenomenon in positive characteristic. They thus measure accurately the algebraic complexity of an equation defining a surface singularity in characteristic ...

Efficient characteristic set methods for computing solutions of a polynomial equation system in a finite field are proposed. We introduce the concept of proper triangular sets and prove that proper triangular sets are square-free and have solutions. We present an improved algorithm which can be used to reduce the zero set of an equation system in general form to the union of zero sets of proper...