14,453 research outputs found

### Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups

The problem of computing \emph{the exponent lattice} which consists of all
the multiplicative relations between the roots of a univariate polynomial has
drawn much attention in the field of computer algebra. As is known, almost all
irreducible polynomials with integer coefficients have only trivial exponent
lattices. However, the algorithms in the literature have difficulty in proving
such triviality for a generic polynomial. In this paper, the relations between
the Galois group (respectively, \emph{the Galois-like groups}) and the
triviality of the exponent lattice of a polynomial are investigated. The
\bbbq\emph{-trivial} pairs, which are at the heart of the relations between
the Galois group and the triviality of the exponent lattice of a polynomial,
are characterized. An effective algorithm is developed to recognize these
pairs. Based on this, a new algorithm is designed to prove the triviality of
the exponent lattice of a generic irreducible polynomial, which considerably
improves a state-of-the-art algorithm of the same type when the polynomial
degree becomes larger. In addition, the concept of the Galois-like groups of a
polynomial is introduced. Some properties of the Galois-like groups are proved
and, more importantly, a sufficient and necessary condition is given for a
polynomial (which is not necessarily irreducible) to have trivial exponent
lattice.Comment: 19 pages,2 figure

### Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral matrices in a norm ball whose
characteristic polynomial has Galois group different than the full symmetric
group is of strictly lower order of magnitude than the number of all such
matrices in the ball, as the radius increases. More generally, we prove a
similar result for the Galois groups associated with elements in any connected
semisimple linear algebraic group defined and simple over a number field $F$.
Our method is based on the abstract large sieve method developed by Kowalski,
and the study of Galois groups via reductions modulo primes developed by Jouve,
Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice
point counting result, and a non-concentration phenomenon for lattice points in
algebraic subvarieties of the group variety, both established previously by the
authors. The results answer a question posed by Rivin and by Jouve, Kowalski
and Zywina, who have considered Galois groups of random products of elements in
algebraic groups.Comment: submitte

### A Proof of Tarskiās Fixed Point Theorem by Application of Galois Connections

Two examples of Galois connections and their dual forms are considered. One
of them is applied to formulate a criterion when a given subset of a complete lattice forms
a complete lattice. The second, closely related to the first, is used to prove in a short way
the Knaster-Tarskiās fixed point theore

### Galois lattice theory for probabilistic visual landmarks

This paper presents an original application of the Galois lattice theory, the visual landmark selection for topological localization of an autonomous mobile robot, equipped with a color camera. First, visual landmarks have to be selected in order to characterize a structural environment. Second, such landmarks have to be detected and updated for localization. These landmarks are combinations of attributes, and the selection process is done through a Galois lattice. This paper exposes the landmark selection process and focuses on probabilistic landmarks, which give the robot thorough information on how to locate itself. As a result, landmarks are no longer binary, but probabilistic. The full process of using such landmarks is described in this paper and validated through a robotics experiment

### Local epsilon isomorphisms

In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya
and Kato for a particular class of Galois modules obtained by tensoring a
Zp-lattice in a crystalline representation of the Galois group of Qp with a
representation of an abelian quotient of the Galois group with values in a
suitable p-adic local ring. This can be regarded as a local analogue of the
Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending
earlier work of Benois and Berger for the cyclotomic extension. We show that
such an epsilon-isomorphism can be constructed using the Perrin-Riou regulator
map, or its extension to the 2-variable case due to the first and third
authors

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