Classification of all the minimal bilinear algorithms for computing the coefficients of the product of two polynomials modulo a polynomial, part I: The algebra G[u]<Q(u)l>, l>1

AbstractIn this paper we will classify all the minimal bilinear algorithms for computing the coefficients of(∑i=0n-1xiui)(∑i=0n-1yiui) mod Q(u)l where deg Q(u)=j,jl=n and Q(u) is irreducible.The case where l=1 was studied in [1]. For l>1 the main results are that we have to distinguish between two cases: j>1 and j=1. The first case is discussed here while the second is classified in [4]. For j>1 it is shown that up to equivalence every minimal (2n-1 multiplications) bilinear algorithm for computing the coefficients of (∑i=0n-1xiui)(∑i=0n-1yiui) mod Q(u)l is done by first computing the coefficients of (∑i=0n-1xiui)(∑i=0n-1yiui) and then reducing it modulo Q(u)l (similar to the case l = 1, [1])

The computational complexity of the Chow form

We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.Comment: 60 pages, Latex2

Unitary and Euclidean representations of a quiver

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension