655 research outputs found
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
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem
The extended L\"uroth's Theorem says that if the transcendence degree of
\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK is 1 then there exists f \in
\KK(\underline{X}) such that \KK(\mathsf{f}_1,\dots,\mathsf{f}_m) is equal
to \KK(f). In this paper we show how to compute with a probabilistic
algorithm. We also describe a probabilistic and a deterministic algorithm for
the decomposition of multivariate rational functions. The probabilistic
algorithms proposed in this paper are softly optimal when is fixed and
tends to infinity. We also give an indecomposability test based on gcd
computations and Newton's polytope. In the last section, we show that we get a
polynomial time algorithm, with a minor modification in the exponential time
decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001
Factorizations of certain bivariate polynomials
We determine the factorization of X*f(X)-Y*g(Y) over K[X,Y] for all
squarefree additive polynomials f,g in K[X] and all fields K of odd
characteristic. This answers a question of Kaloyan Slavov, who needed these
factorizations in connection with an algebraic-geometric analogue of the Kakeya
problem.Comment: 5 page
Pseudo-modularity and Iwasawa theory
We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is
Gorenstein at an intersection point between the Eisenstein family and the
cuspidal locus. As a corollary, we obtain new results on Sharifi's conjecture.
This result is achieved by constructing a universal ordinary pseudodeformation
ring and proving an result.Comment: Changes to section 5.9; typos corrected. To appear in Amer. J. Math.
54 page
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