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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
Splitting Behavior of -Polynomials
We analyze the probability that, for a fixed finite set of primes S, a
random, monic, degree n polynomial f(x) with integer coefficients in a box of
side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with
splitting field over the rationals having Galois group ; (ii) the
polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii)
f(x) has a prescribed splitting type at each prime p in S.
The limit probabilities as are described in terms of values of
a one-parameter family of measures on , called splitting measures, with
parameter evaluated at the primes p in S. We study properties of these
measures. We deduce that there exist degree n extensions of the rationals with
Galois closure having Galois group with a given finite set of primes S
having given Artin symbols, with some restrictions on allowed Artin symbols for
p<n. We compare the distributions of these measures with distributions
formulated by Bhargava for splitting probabilities for a fixed prime in
such degree extensions ordered by size of discriminant, conditioned to be
relatively prime to .Comment: 33 pages, v2 34 pages, introduction revise
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