Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.Comment: to appear in SIAM Journal on Discrete Mathematic

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

On some classes of irreducible polynomials over finite fields

In this thesis we describe the work in literature on various aspects of the theory of polynomials over nite elds. We focus on properties like irreducibility and divisibility. We also consider existence and enumeration problems for irreducible polynomials of special types. After the introductory Chapter 1, we collect the well-known results on irreducibility of binomials and trinomials in Chapter 2, where we also present the number of irreducible factors of a xed degree k of xt due to L. Redei. Chapter 3 is on self-reciprocal polynomials. An in nite family of irreducible, self-reciprocal polynomials over F2 is presented in Section 3.2. This family was obtained by J. L. Yucas and G. L. Mullen. Divisibility of self-reciprocal polynomials over F2 and F3 is studied in Sections 3.3 and 3.4 following the work of R. Kim and W. Koepf. The last chapter aims to give a survey of recent results concerning existence and enumeration of irreducible polynomials with prescribed coefficients

Prime and composite Laurent polynomials

In 1922 Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case when f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.Comment: Some of the proofs given in sections 6-8 are simplified. Some other small alterations were mad