Output-sensitive modular algorithms for polynomial matrix normal forms

Output-sensitive modular algorithms for polynomial matrix normal forms

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorit...

Parallel Algorithms for Matrix Normal Forms

Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the input matrix are computed, and the algorithm is probabilistic of Las Vegas type, i.e., always finds t...

Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix

Improved output-sensitive quantum algorithms for Boolean matrix multiplication

We present new quantum algorithms for Boolean Matrix Multiplication in both the time complexity and the query complexity settings. As far as time complexity is concerned, our results show that the product of two n× n Boolean matrices can be computed on a quantum computer in time Õ(n3/2+nl3/4), where l is the number of non-zero entries in the product, improving over the outputsensitive quantum a...

Stable normal forms for polynomial system solving

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal I . The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [29]. This general border basis algorithm weakens the monomial ordering requirement for Gröbner bases computations. It is up to date the ...