3,039 research outputs found
Number of Irreducible Polynomials and Pairs of Relatively Prime Polynomials in Several Variables over Finite Fields
We discuss several enumerative results for irreducible polynomials of a given
degree and pairs of relatively prime polynomials of given degrees in several
variables over finite fields. Two notions of degree, the {\em total degree} and
the {\em vector degree}, are considered. We show that the number of
irreducibles can be computed recursively by degree and that the number of
relatively prime pairs can be expressed in terms of the number of irreducibles.
We also obtain asymptotic formulas for the number of irreducibles and the
number of relatively prime pairs. The asymptotic formulas for the number of
irreducibles generalize and improve several previous results by Carlitz, Cohen
and Bodin.Comment: 33 page
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
Generating series for irreducible polynomials over finite fields
We count the number of irreducible polynomials in several variables of a
given degree over a finite field. The results are expressed in terms of a
generating series, an exact formula and an asymptotic approximation. We also
consider the case of the multi-degree and the case of indecomposable
polynomials
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
Tuples of polynomials over finite fields with pairwise coprimality conditions
Let q be a prime power. We estimate the number of tuples of degree
bounded monic polynomials (Q1, . . . , Qv) ∈ (Fq[z])v that satisfy given
pairwise coprimality conditions. We show how this generalises from monic
polynomials in finite fields to Dedekind domains with a finite norm
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
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