Filters

169,112 Hits in 4.3 sec

### Factoring polynomials over global fields I

Michael E. Pohst
2005 Journal of symbolic computation
In this paper we present a generic algorithm for factoring polynomials over global fields F.  ...  Generic Factorization Algorithm. Input An integral domain R with quotient field K and a square-free polynomial g(t) ∈ K [t] of degree greater than one. Step 1.  ...  Factorizations of g(t) over R and over K can differ substantially. For global fields the concept of algebraic integers can be used to solve this problem.  ...

### Sentences over Integral Domains and Their Computational Complexities

Shih Ping Tung
1999 Information and Computation
As is well known, for all global fields K there also are algorithms for factoring polynomials over K (Fried and Jarden, 1986) .  ...  Factoring polynomials over K with parameters \_ sentences Ring of integers of a global field Co-NP-complete _\ sentences Ring of integers of a global field NP-complete \_ sentences K[T] :  ...

### On invariance of degree for certain computations

Michael Maller
2004 Journal of Complexity
a polynomial of degree n: This generalizes results of Cucker, and holds for polynomials and machines over an infinite ordered topological field and in certain cases over Z as well.  ...  We prove that a sequential or parallel machine in the Blum-Shub-Smale model, which recognizes the roots of an irreducible polynomial f ðx; yÞ of degree n; in globally bounded time T; must actually compute  ...  Let k be a field and f i Ak½x 1 ; x 2 ; yx n ; i ¼ 1; y; m; be polynomials in n variables, V ð f Þ ¼ fxAk n j f ðxÞ ¼ 0g and V ðff i j 1pipmgÞ ¼ fxAk n j f i ðxÞ ¼ 0; i ¼ 1ymg: Proof.  ...

### Rational separability over a global field

Alexandra Shlapentokh
1996 Annals of Pure and Applied Logic
F such that for every x E RI for some i, Ci(x) E R2.  ...  elements of R2 do not have factors of relative degree 1 in some simple extension of K.  ...  Lemma 3 . 1 . 31 Let K be a global field and let T(x) = Cik,,aix', where ai E K. Let c = c(T) = i (max lordPail), i=(J P where p ranges over all the non-archimedean primes of K.  ...

### A BSP Parallel Model for the Göttfert Algorithm over F 2 [chapter]

Fatima Abu Salem
2004 Lecture Notes in Computer Science
In Niederreiter's factorization algorithm for univariate polynomials over finite fields, the factorization problem is reduced to solving a linear system over the finite field in question, and the solutions  ...  are used to produce the complete factorization of the polynomial into irreducibles.  ...  Let f be a polynomial of degree d over F 2 , and f = g e1 1 ...g em m be its canonical factorization of over the field. Let N f be the Niederreiter matrix of coefficients of f ([1], [2] ).  ...

### Irreducibility of polynomials over global fields is diophantine

Philip Dittmann
2018 Compositio Mathematica
Given a global field \$K\$ and a positive integer \$n\$ , we present a diophantine criterion for a polynomial in one variable of degree \$n\$ over \$K\$ not to have a root in \$K\$ .  ...  We also deduce a diophantine criterion for a polynomial over \$K\$ of given degree in a given number of variables to be irreducible.  ...  I would like to thank my supervisor, Jochen Koenigsmann, for helpful discussions on the subject of this paper and comments on the exposition.  ...

### Factoring polynomials over global fields II

José Méndez Omaña, Michael E. Pohst
2005 Journal of symbolic computation
Also, a generalization of the application of LLL reduction for factoring polynomials over arbitrary global fields is developed.  ...  In this paper we describe software for an efficient factorization of polynomials over global fields F. The algorithm for function fields was recently incorporated into our system KANT.  ...  The factorization of polynomials over finite fields in Step 2 is not discussed in this paper.  ...

### Computing the Galois group of a polynomial over a p-adic field [article]

Christopher Doris
2020 arXiv   pre-print
We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the "naive" algorithm, these are the first general algorithms for this task.  ...  Similarly a global model for F (x) ∈ K[x] extending i is k F k where F = k F k is the factorization over K of F into irreducible factors, L k /K are the corresponding extensions, i k : L k → L k are global  ...  Factors This factorizes F (x) = k F k (x) into irreducible factors over K, produces a global model F k (x) for each factor, and then the global model is F(x) = k F k (x).  ...

### Finite fields and applications: Proceedings of the third international conference, Glasgow, July 1995

1997 Computers and Mathematics with Applications
Factoring cyclotomic polynomials over large finite fields (Greg Stein). Character sums and coding theory (Serguei A. Stepanov).  ...  Cellular automata, substitutions and factorization of polynomials over finite fields (Ehler Lange). A new class of two weight codes (Philippe Langevin).  ...

### Modern compiler implementation in ML: Basic techniques

1997 Computers and Mathematics with Applications
Factoring cyclotomic polynomials over large finite fields (Greg Stein). Character sums and coding theory (Serguei A. Stepanov).  ...  Cellular automata, substitutions and factorization of polynomials over finite fields (Ehler Lange). A new class of two weight codes (Philippe Langevin).  ...

### Java language reference

1997 Computers and Mathematics with Applications
Factoring cyclotomic polynomials over large finite fields (Greg Stein). Character sums and coding theory (Serguei A. Stepanov).  ...  Cellular automata, substitutions and factorization of polynomials over finite fields (Ehler Lange). A new class of two weight codes (Philippe Langevin).  ...

### Modern compiler implementation in C: Basic techniques

1997 Computers and Mathematics with Applications
Factoring cyclotomic polynomials over large finite fields (Greg Stein). Character sums and coding theory (Serguei A. Stepanov).  ...  Cellular automata, substitutions and factorization of polynomials over finite fields (Ehler Lange). A new class of two weight codes (Philippe Langevin).  ...

### Modern compiler implementation in Java: Basic techniques

1997 Computers and Mathematics with Applications
Factoring cyclotomic polynomials over large finite fields (Greg Stein). Character sums and coding theory (Serguei A. Stepanov).  ...  Cellular automata, substitutions and factorization of polynomials over finite fields (Ehler Lange). A new class of two weight codes (Philippe Langevin).  ...

### Characteristic polynomials of central simple algebras [article]

Chia-Fu Yu
2012 arXiv   pre-print
We characterize characteristic polynomials of elements in a central simple algebra.  ...  We also give an account for the theory of rational canonical forms for separable linear transformations over a central division algebra, and a description of separable conjugacy classes of the multiplicative  ...  When F is a global field, these can be computed explicitly using Brauer groups over local fields [6] and the period-index relation [ Which polynomial is characteristic?  ...