Counting decomposable univariate polynomials

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker

Survey on counting special types of polynomials

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f is decomposable if f = g o h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f. We present a classification of all collisions at degree n = p^2 which yields an exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann (editors), Computer Algebra and Polynomials, Lecture Notes in Computer Scienc

Fast Arithmetics Using Chinese Remaindering

In this paper, some issues concerning the Chinese remaindering representation are discussed. Some new converting methods, including an efficient probabilistic algorithm based on a recent result of von zur Gathen and Shparlinski \cite{Gathen-Shparlinski}, are described. An efficient refinement of the NC$^1$ division algorithm of Chiu, Davida and Litow \cite{Chiu-Davida-Litow} is given, where the number of moduli is reduced by a factor of $\log n$

Permanent and determinant

AbstractThe n×n permanent is not a projection of the m×m determinant if m ⩽ √2n− 6√n