SIGSAM

In [7] we established a Picard-Vessiot theory over differentially simple rings which may not be fields. Differential modules over such rings were proven to be locally free but don't have to be free as modules. In this article, we give a family of ...

In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form x^α − x^β = (α − β)vx^α+β. In the paper, he studied the series solution of this equation and other results based on an assumption which was not proved ...

Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the ...

The work reported here is motivated by problems arising in solving polynomial systems over the real numbers.

Evaluation of the Hessian matrix of a scalar function is a subproblem in many numerical optimization algorithms. For large-scale problems often the Hessian matrix is sparse and structured, and it is preferable to exploit such information when available. ...

We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by q₀(z) = 0, q₁(z) = 1, and q_n(z) = zq_n−1q_n−2 + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two ...

Bohemian eigenvalues are the eigenvalues of matrices with entries of bounded height, typically drawn from a discrete set. We will call this set F with cardinality #F. The name "Bohemian" is intended as a mnemonic and is derived from "bounded height ...

In his pioneering work [1, 2], Michael Karr introduced ΠΣ-fields which provide a rather general framework for symbolic summation. He worked out the first algorithmic steps to represent indefinite nested sums and products as transcendental extensions ...

Wiedemann's paper, introducing his algorithm for sparse and structured matrix computations over arbitrary fields, also presented a pair of matrix preconditioners for computations over small fields. The analysis of the second of these is extended here in ...

Polynomials eigenvalue problems with structured matrix polynomials arise in many applications. The standard way to solve polynomial eigenvalue problems is through the classical Frobenius companion linearizations, which may not retain the structure of ...

Let x = (x₁,...,x_n) ∈ R ⁿ and λ ∈ R. A smooth map f(x,λ) is called Z₂-equivariant (Z₂-invariant) if f(−x, λ) = −f(x,λ) (f(−x, λ) = f(x,λ)). Consider the local solutions of a Z₂-equivariant map f(x,λ) = 0 around a solution, say f(x₀,λ₀), as the ...

The exact probability, dependent on the matrix structure, is given that the block Wiedemann algorithm correctly computes the leading invariant factors of a matrix. A tight lower bound, structure independent, is derived.

The ISCZ method (Interval-Symbol method with Correct Zero rewriting) was proposed in [2] based on Shirayanagi-Sweedler stabilization theory ([3]), to reduce the amount of exact computations as much as possible to obtain the exact results. The authors of ...

We deal with Temperley-Lieb algebras of type B, extending the result in [3, §6]. By completing the relations coming from a presentation of the Temperley-Lieb algebra, we find its Gröbner-Shirshov basis to obtain the corresponding set of standard ...

We are interested in specific structured matrices obtained from [5] which arise from combinatorial problems.

Let (K, σ) be a difference field. We define the set of constants by const K = {c ∈ K | σ(c) = c}. A ΠΣ*-extension of K is a field of rational functions K(t) over K together with an extension of σ to K(t) given by either σ(t) = at (Π case) or σ(t) = t + ...

We employ two techniques to dramatically improve Maple's performance on the Fermat benchmarks for simplifying rational expressions. First, we factor expanded polynomials to ensure that gcds are identified and cancelled automatically. Second, we replace ...

It is with great sadness that we annonce you the death of our friend and colleague Marc Rybowicz. He passed away on November 11th after a long fight against cancer since June 2015.