A new modular algorithm for computation of algebraic number polynomial gcds
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
The computation of polynomial greatest common divisors over an algebraic number field
Journal of Symbolic Computation
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
Algorithmic algebra
A course in computational algebraic number theory
A course in computational algebraic number theory
Factoring Polynomials Over Algebraic Number Fields
ACM Transactions on Mathematical Software (TOMS)
AAECC-8 Proceedings of the 8th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Factoring polynominals over algebraic number fields
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
A p-adic algorithm for univariate partial fractions
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
Algorithms for polynomials over a real algebraic number field.
Algorithms for polynomials over a real algebraic number field.
On the computation of integral bases and defects of integrity
On the computation of integral bases and defects of integrity
Improvements in cad-based quantifier elimination
Improvements in cad-based quantifier elimination
P-adic reconstruction of rational numbers
ACM SIGSAM Bulletin
Minimum converging precision of the QR-factorization algorithm for real polynomial GCD
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
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Modular methods for computing the gcd of two univariate polynomials over an algebraic number field require a priori knowledge about the denominators of the rational numbers in the representation of the gcd. We derive a multiplicative bound for these denominators without assuming that the number generating the field is an algebraic integer. Consequently, the gcd algorithm of Langemyr and McCallum [J. Symbolic Computation, 8:429-448, 1989] can now be applied directly to polynomials that are not necessarily represented in terms of an algebraic integer. Worst-case analyses and experiments with an implementation show that by avoiding a conversion of representation the reduction in the computing time can be significant. We also suggest the use of an algorithm for recovering a rational number from its modular residue so that the denominator bound need not be computed explicitly. Experiments and analyses indicate that this is a good practical alternative.