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

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

A note on Gao’s algorithm for polynomial factorization

AbstractShuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f

Counting reducible and singular bivariate polynomials

AbstractAmong the bivariate polynomials over a finite field, most are irreducible. We count some classes of special polynomials, namely the reducible ones, those with a square factor, the “relatively irreducible” ones which are irreducible but factor over an extension field, and the singular ones, which have a root at which both partial derivatives vanish

Factoring Polynomials and Groebner Bases

Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied in the last few decades by many mathematicians and computer scientists. The main approaches include Berlekamp\u27s method (1967) based on the kernel of Frobenius map, Niederreiter\u27s method (1993) via an ordinary differential equation, Zassenhaus\u27s modular approach (1969), Lenstra, Lenstra and Lovasz\u27s lattice reduction (1982), and Gao\u27s method via a partial differential equation (2003). These methods and their recent improvements due to van Hoeij (2002) and Lecerf et al (2006--2007) provide efficient algorithms that are widely used in practice today. This thesis studies two issues on polynomial factorization. One is to improve the efficiency of modular approach for factoring bivariate polynomials over finite fields. The usual modular approach first solves a modular linear equation (from Berlekamp\u27s equation or Niederreiter\u27s differential equation), then performs Hensel lifting of modular factors, and finally finds right combinations. An alternative method is presented in this thesis that performs Hensel lifting at the linear algebra stage instead of lifting modular factors. In this way, there is no need to find the right combinations of modular factors, and it finds instead the right linear space from which the irreducible factors can be computed via gcd. The main advantage of this method is that extra solutions can be eliminated at the early stage of computation, so improving on previous Hensel lifting methods. Another issue is about whether random numbers are essential in designing efficient algorithms for factoring polynomials. Although polynomials can be quickly factored by randomized polynomial time algorithms in practice, it is still an open problem whether there exists any deterministic polynomial time algorithm, even if generalized Riemann hypothesis (GRH) is assumed. The deterministic complexity of factoring polynomials is studied here from a different point of view that is more geometric and combinatorial in nature. Tools used include Gr\u27{o}bner basis structure theory and graphs, with connections to combinatorial designs. It is shown how to compute deterministically new Gr\u27{o}bner bases from given G\u27{o}bner bases when new polynomials are added, with running time polynomial in the degree of the original ideals. Also, a new upper bound is given on the number of ring extensions needed for finding proper factors, improving on previous results of Evdokimov (1994) and Ivanyos, Karpinski and Saxena (2008)

Hensel lifting and bivariate polynomial factorisation over finite fields

Abstract. This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials. 1

Hensel Lifting And Bivariate Polynomial Factorisation Over Finite Fields

. This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials. 1. Introduction It is well known that the Hensel lifting technique provides practical methods for factoring polynomials over various fields. Such methods are known to run in exponential time in the worst case, but seem fast for most polynomials. The latter phenomenon has not been fully understood and calls for an average running time analysis. The only analysis we know of is that of Collins 1979 [4] for univariate integral polynomials (factoring over the rational numbers). He shows, under some reasonable number theoretic conjectures, that the average running time is indeed polynomial. In this paper, we present a rigorous analysis for bivariate polynomials over finite fields. We sh..