63,860 research outputs found
Computing and Using Minimal Polynomials
Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a “resolved problem”. But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality
Computing and Using Minimal Polynomials
Given a zero-dimensional ideal I in a polynomial ring, many computations
start by finding univariate polynomials in I. Searching for a univariate
polynomial in I is a particular case of considering the minimal polynomial of
an element in P/I. It is well known that minimal polynomials may be computed
via elimination, therefore this is considered to be a "resolved problem". But
being the key of so many computations, it is worth investigating its meaning,
its optimization, its applications
On Sequences, Rational Functions and Decomposition
Our overall goal is to unify and extend some results in the literature
related to the approximation of generating functions of finite and infinite
sequences over a field by rational functions. In our approach, numerators play
a significant role. We revisit a theorem of Niederreiter on (i) linear
complexities and (ii) ' minimal polynomials' of an infinite sequence,
proved using partial quotients. We prove (i) and its converse from first
principles and generalise (ii) to rational functions where the denominator need
not have minimal degree. We prove (ii) in two parts: firstly for geometric
sequences and then for sequences with a jump in linear complexity. The basic
idea is to decompose the denominator as a sum of polynomial multiples of two
polynomials of minimal degree; there is a similar decomposition for the
numerators. The decomposition is unique when the denominator has degree at most
the length of the sequence. The proof also applies to rational functions
related to finite sequences, generalising a result of Massey. We give a number
of applications to rational functions associated to sequences.Comment: Several more typos corrected. To appear in J. Applied Algebra in
Engineering, Communication and Computing. The final publication version is
available at Springer via http://dx.doi.org/10.1007/s00200-015-0256-
Gauss periods are minimal polynomials for totally real cyclic fields of prime degree
We report extensive computational evidence that Gauss period equations are
minimal discriminant polynomials for primitive elements representing Abelian
(cyclic) polynomials of prime degrees . By computing 200 period equations up
to , we significantly extend tables in the compendious number fields
database of Kl\"uners and Malle. Up to , period equations reproduce known
results proved to have minimum discriminant. For , period
equations coincide with 53 known but unproved cases of minimum discriminant in
the database, and fill a gap of 19 missing cases. For , we
report 128 not previously known cases, 16 of them conjectured to be minimum
discriminant polynomials of Galois group . The significant advantage of
period equations is that they all may be obtained analytically using a
procedure that works for fields of arbitrary degrees, and which are extremely
hard to detect by systematic numerical search.Comment: 7 pages, 4 tables, no figure
Computing Minimal Polynomials of Matrices
We present and analyse a Monte-Carlo algorithm to compute the minimal
polynomial of an matrix over a finite field that requires
field operations and O(n) random vectors, and is well suited for successful
practical implementation. The algorithm, and its complexity analysis, use
standard algorithms for polynomial and matrix operations. We compare features
of the algorithm with several other algorithms in the literature. In addition
we present a deterministic verification procedure which is similarly efficient
in most cases but has a worst-case complexity of . Finally, we report
the results of practical experiments with an implementation of our algorithms
in comparison with the current algorithms in the {\sf GAP} library
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