237 research outputs found
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
The number of reducible space curves over a finite field
"Most" hypersurfaces in projective space are irreducible, and rather precise
estimates are known for the probability that a random hypersurface over a
finite field is reducible. This paper considers the parametrization of space
curves by the appropriate Chow variety, and provides bounds on the probability
that a random curve over a finite field is reducible
Counting classes of special polynomials
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. In two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. We present counting results 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). These numbers come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f over a field F is decomposable if f = g o h with 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 F does not divide n = deg f, is fairly well understood, and the upper and lower bounds on the number of decomposable polynomials of degree n match asymptotically. 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. There is an obvious inclusion-exclusion formula for counting. The main issue is then to determine, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs. We provide a normal form for collisions of any number of compositions with any number of components. This generalization yields an exact formula for the number of decomposable polynomials of degree n coprime to p. For the wild case, we classify all collisions at degree n = p^2 and obtain the exact number of decomposable polynomials of degree p^2
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
Liminal reciprocity and factorization statistics
Let denote the number of monic irreducible polynomials in
of degree . We show that for a fixed
degree , the sequence converges -adically to an explicitly
determined rational function . Furthermore we show that the
limit is related to the classic necklace polynomial
by an involutive functional equation, leading to a phenomenon we
call liminal reciprocity. The limiting first moments of factorization
statistics for squarefree polynomials are expressed in terms of a family of
symmetric group representations as a consequence of liminal reciprocity.Comment: 22 pages. To appear in Algebraic Combinatoric
Algorithmic Algebraic Geometry and Flux Vacua
We develop a new and efficient method to systematically analyse four
dimensional effective supergravities which descend from flux compactifications.
The issue of finding vacua of such systems, both supersymmetric and
non-supersymmetric, is mapped into a problem in computational algebraic
geometry. Using recent developments in computer algebra, the problem can then
be rapidly dealt with in a completely algorithmic fashion. Two main results are
(1) a procedure for calculating constraints which the flux parameters must
satisfy in these models if any given type of vacuum is to exist; (2) a stepwise
process for finding all of the isolated vacua of such systems and their
physical properties. We illustrate our discussion with several concrete
examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure
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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