Orthogonal Sequences of Polynomials over Arbitrary Fields

Multiplication over Arbitrary Fields

We prove a lower bound of 52n2 3n for the rank of n n–matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.

On Fast Multiplication of Polynomials Over Arbitrary

Fast Multiplication of Polynomials over Arbitrary Rings*

An algorithm is presented that allows to multiply two univariate polynomials of degree no more than n with coefficients from an arbitrary (possibly non-commutative) ring in O(n log(n) log(log n)) additions and subtractions and O(n log(n)) multiplications. The arithmetic depth of the algorithm is O(log(n)). This algorithm is a modification of the Schönhage-Strassen procedure to arbitrary radix f...

Small Oscillations, Sturm Sequences, and Orthogonal Polynomials

The relation between small oscillations of one-dimensional mechanical «-particle systems and the theory of orthogonal polynomials is investigated. It is shown how the polynomials provide a natural tool to determine the eigenfrequencies and eigencoordinates completely, where the existence of a specific two-termed recurrence formula is essential. Physical and mathematical statements are formulate...

Quadratic Forms over Arbitrary Fields

Introduction. Witt [5] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only...